STRONG SOLUTIONS FOR SEMILINEAR WAVE EQUATIONS WITH ...

STRONG SOLUTIONS FOR SEMILINEAR WAVE EQUATIONS WITH DAMPING AND SOURCE TERMS PETRONELA RADU

Abstract. This paper deals with local existence of strong solutions for semilinear wave equations with power-like interior damping and source terms. A long standing restriction on the range of exponents for the two nonlinearities governs the literature on wellposedness of weak solutions of finite energy. We show that this restriction may be eliminated for the existence of higher regularity solutions by employing natural methods that use the physics of the problem. This approach applies to the Cauchy problem posed on the entire Rn as well as for initial boundary problems with homogeneous Dirichlet boundary conditions. Keywords: wave equation, local existence, finite speed of propagation, nonlinear damping, interior source, strong solutions 2000 MSC: Primary: 35L05; Secondary: 35L20

1. Introduction 1.1. Description of the model. Consider the following semilinear wave equation which exhibits a competitive interaction between a nonlinear source and a damping term:   utt − ∆u + f (u) + g(ut ) = 0, (x, t) ∈ Ω × [0, ∞) (SW) (u, ut )|t=0 = (u0 , u1 ), x ∈ Ω  u = 0, (x, t) ∈ ∂Ω × (0, T ). The domain Ω ⊂ Rn may be a bounded or unbounded regular domain in Rn (possibly the entire Rn ), for 3 ≤ n < 6. The focus here is on strong solutions (u, ut ) ∈ H 2 × H 1 for which we show local in time existence. The nonlinear terms are usually thought of as being powers of u, respectively ut , with a prototype equation given by: utt − ∆u + au|u|p−1 + but |ut |m−1 = 0. Thus, a = 0, b = 1 yields the nonlinear damped wave equation, while for a = 1, b = 0 we obtain the nonlinear Klein-Gordon equation. For our purposes, the more interesting case to be analyzed is a < 0 and b > 0 when the two nonlinearities compete with each other. The considerable interest that the system has received over the past decade is attributed mainly to the interaction between the two nonlinearities. In other models the nonlinearities appear as boundary conditions, instead of affecting the system internally; this brings in Date: August 9, 2011. 1

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PETRONELA RADU

other interesting (and often) technical aspects, but regardless of where the terms arise (internally or on the boundary), the crux of the problem lies in the question: Does the energy lost through damping compensate for the energy brought into the system by f , thus extending the lifespan of the solution? The intuition is that an increase in energy will lead to blow-up of solutions in finite time, while a decrease would lead to global existence of solutions (unless the system experiences overdamping). Roughly speaking, g extends the life span of the solution while taking energy out of the system, whereas f has the opposite effect. Mathematically, it has been shown (for certain ranges of exponents p and m) that the interplay between these two power-like nonlinearities is exhibited through their exponents. Thus, if the exponent p of the energy accretive source f (u) = −|u|p−1 u is higher than the exponent m of the damping term g(ut ) = ut |ut |m−1 , then solutions cease to exist after a finite time, whereas if the exponent of the damping term dominates the interaction, then the system will have a solution for arbitrarily large times ([8], [5]). Thus, by juxtaposing the two nonlinear terms to the wave operator one expects a system where the issue of existence vacillates between global existence and blow up in finite time. The blow-up phenomenon occurs when “bad” sources pump energy into the system, so the norm of the solution becomes unbounded in finite time. For “good” sources (e.g., f (u) = |u|p−1 u) global existence of solutions is expected in the subcritical case p < 5 in n = 3 dimensions (see [13]); for supercritical exponents p > 5 see the discussion on Lebeau’s ill-posedness results below. The interaction of the nonlinearities plays the determinant role not only for the analysis of solutions for large times, but it is foremost critical for the local existence of solutions, as we demonstrate in this paper. For certain ranges of exponents p and m the local (in time) existence of weak solutions remains an open problem. The following restriction for the range of exponents p and m governed previous existence results concerning (SW) (R)

p+

p 2n < m n−2

for

n n+2 n−2 . The theory of Orlicz spaces was employed in [16] to allow source and damping terms that satisfy the restriction (R). These results are obtained for the Cauchy problem on Rn . The author showed that local existence holds on bounded domains with homogeneous Dirichlet data in [14], while the works [3, 4, 5, 6] extended the results further to supercritical sources 5 < p < 6 in n = 3 on bounded and unbounded domains. Our approach here uses the patching argument first used by Tartar and Crandall in [17, 18] for the Broadwell model, and later redeveloped by the author for semilinear wave equations in [13, 14]. This method turns out to be an extremely valuable tool that has great potential for many problems for which some energy (or “entropy”) estimates are available, and for which a finite speed of propagation property holds. Eliminating the restriction (R) for strong solutions requires additional work, such as the derivation of higher order estimates.

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PETRONELA RADU

Once these higher order estimates are obtained we employ a second order potential well argument to obtained bounds on the higher order derivatives of the solution. This idea is complemented by the robustness of the patching argument to yield local existence of strong solutions. The methods used are natural from a physical point of view and they apply to the investigation of problems on both, bounded and unbounded domains, in contrast with other methods that are available and which apply exclusively to only one of the setups. Also, earlier works [2, 8, 16] used the smoothing effect of the damping term, but our method circumvents this step. By making stronger use of the monotonicity method we are able to show local wellposedness for the problem which contains only source terms (g = 0 or m = 0). Finally, we remark that our results hold for finite energy initial data, not necessarily with compact support as it is the case in [16, 19]. Note that although we will focus on the Cauchy problem associated with this semilinear wave equation, the wellposedness also holds on a bounded domain. The applicability of the method to the boundary value problems was shown in [14], but for the range of exponents described by the restriction (R). We will only sketch the necessary adjustments, without going through the complete arguments which follow closely [14]. Regarding the behavior of solutions to this problem a few more facts should be pointed out in order to give a better picture of existence of regular solutions. First, one may not be guar2n anteed even local existence of strong solutions for p ≥ n−2 as Lebeau showed in [9, 10] that 2 for odd supercritical exponents the problem is ill-posed in all spaces H s for s ∈ (1, n2 − p−1 ) p−1 when f (u) = |u| u and g(ut ) = 0 (note that this is the case of a “good” absorption source term so the energy of the system decreases). He constructed solutions that blowup immediately in these H s norms. However, in the presence of some special damping terms, positive results regarding global existence of solutions have been established. Thus, some preliminary results [20] seem to indicate global existence of H 2 solutions in the presence of arbitrarily large polynomial absorption sources and cubic damping term g(ut ) = u3t .

For easy navigation through the paper we provide here a brief outline. The setup of the problem is done in the next section where we specify notations, conditions satisfied by the problem, and recall the definitions for weak and strong solutions. The proof is built on results (most of them classical) for the problem with Lipschitz source terms. These results are the energy identity, global existence of solutions, and two versions of the finite speed of propagation property; they are presented in section 3 where we also provide references for the proofs. The main theorem which gives local existence of strong solutions is presented in section 4 with the main steps organized in subsections. We conclude the paper with some final remarks and conclusions. Many arguments of the proof coincide with arguments used in [13, 14] but we reproduce only the essential ones here.

STRONG SOLUTIONS FOR SEMILINEAR WAVE EQUATIONS

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2. Notation, assumptions, definitions In the sequel we use the following notation: | · |q,Ω is the norm in Lq (Ω). For q = 2 we take the liberty to drop the subscript 2, so | · |Ω is the norm in L2 (Ω), while for Ω = Rn we drop the subscript Ω; thus, | · |q denotes the norm in Lq (Rn ). The source and damping terms satisfy the following assumptions: (Af ) f ∈ C 2 (R), f (0) = f 0 (0) = 0, and there exist f1 , f2 > 0 such that: n+2 00 p−2 |f (u)| ≤ max{f1 |u| , f2 }, for some p ∈ 1, . n−2 (Ag ) g ∈ C 1 (R), g(0) = 0, g 0 (0) ≥ 0, and g satisfies the growth assumptions: vg(v) ≥ g1 |v|m+1 ,

|g(v)| ≤ g2 |v|m ,

for some m ≥ 0 and g1 , g2 > 0. Remark 2.1.RAs a consequence of (Af ) we have the following growth conditions for f (u) u and F (u) = 0 f (y)dy: |f (u)| ≤ max{f1 |u|p , f2 |u|2 }, f1 f2 |F (u)| ≤ max{ |u|p+1 , |u|3 }, p+1 3

(2.1)

n+2 for p ∈ 1, n−2 . We also make the following convention. The problem (SW) will be denoted by (SWB) when we specifically impose that Ω is a bounded domain. Whenever the problem is posed on an arbitrary domain that is not required to be bounded (possibly Rn ) we will use the label (SW). The meanings of weak and strong solutions for (SW) are classical, but we include the two definitions below for clarity and preciseness reasons. Definition 2.2 (Weak solutions). Let ΩT := Ω×(0, T ), where Ω ⊂ Rn is an open connected set with smooth boundary ∂Ω. Suppose the functions f and g satisfy the assumptions (Af ) and (Ag ), and further suppose that u0 ∈ H01 (Ω) ∩ Lp+1 (Ω) and u1 ∈ L2 (Ω) ∩ Lm+1 (Ω). A weak solution on ΩT of the boundary value problem (SW) is any function u satisfying u ∈ C(0, T ; H01 (Ω)) ∩ Lp+1 (ΩT ), and Z ΩT

ut ∈ L2 (ΩT ) ∩ Lm+1 (ΩT ),

u(x, s)φtt (x, s) + ∇u(x, s) · ∇φ(x, s) + f (u)φ(x, s) Z + g(ut )φ(x, s) dxds = u1 (x)φ(x, 0) − u0 (x)φt (x, 0) dx Ω

for every φ ∈

Cc∞ (Ω

× (−∞, T )).

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Definition 2.3 (Strong solutions). Let ΩT , f, and g satisfy the assumptions from the above definition. Assume u0 ∈ H 2 (Ω) ∩ H01 (Ω) and u1 ∈ H 1 (Ω) ∩ Lm+1 (Ω). A strong solution u for the problem (SW) is a weak solution in the sense of Definition 2.2 with the additional regularity u ∈ C 1 ([0, T ]; L2 (Ω)) and utt , ∆u ∈ L2 (0, T ; L2 (Ω)). Remark 2.4. The additional regularity required by a strong solution of (SW) as described above implies that (u(·, t), ut (·, t)) ∈ (H 2 (Ω) ∩ H01 )(Ω) × H 1 (Ω) for all t > 0. Finally, we denote by C all constants whose values do not affect the proof. We will specify and keep track only of the constants that are important in our arguments. 3. Preliminary results. Lipschitz source terms and general damping In this section we will recall some results regarding strong solutions for (SW) with Lipschitz source terms and damping when the initial data (u0 , u1 ) ∈ (H 2 ∩ H01 ) × H 1 . Throughout the section we assume that there exists L > 0 such that: (3.1)

|f (u) − f (v)| ≤ L|u − v|,

for all u, v ∈ R. The first result gives us the basic bound for the semilinear wave equation (SW); for a proof see [13]. Proposition 3.1 (The energy identity). Let Ω be a bounded domain in Rn with C 1 boundary, u0 ∈ H 2 (Ω) ∩ H01 (Ω), u1 ∈ H 1 (Ω). If u is a strong solution of (SW), where f, g satisfy (Af ), (Ag ), and (3.1), then we have the following equality: Z Z tZ (3.2) E(t) + F (u(x, t)) dx + g(ut (x, s))ut (x, s) dx ds = E(0), Ω 0 Ω Rv where F (v) = 0 f (y)dy and 1 1 E(t) := |ut (t)|2Ω + |∇u(t)|2Ω . 2 2 Existence of strong solutions in the presence of globally Lipschitz sources is a well known result; see [1] for a proof using monotone semigroup theory, or [13] for a monotonicity argument involving a-priori estimates. Theorem 3.2 (Existence and uniqueness of strong solutions for dissipative wave equations with Lipschitz source terms). Let Ω ⊂ Rn be a bounded domain with C 1 boundary, and let the functions f and g satisfy (Af ) and (Ag ). Additionally, f is Lipschitz. Let u0 ∈ H 2 (Ω) ∩R H01 (Ω), u1 ∈ H01 (Ω) with G(u1 ) ∈ L1 (Ω), where G is defined by the formula v G(v) = 0 g(y)dy. Then for any T > 0 the initial boundary-value problem:   utt − ∆u + f (u) + g(ut ) = 0 in Ω × (0, T ); (u, ut )|t=0 = (u0 , u1 ) in Ω; (SWB)  u = 0 on ∂Ω × (0, T ).


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admits a unique strong solution u on the time interval [0, T ] in the sense of Definition 2.3. Remark 3.3. By using monotone semigroup theory one can prove that this theorem holds for (u0 , u1 ) ∈ [H k (Ω) ∩ H01 (Ω)] × H k−1 (Ω) which generate a solution u ∈ H k (Ω) ∩ H01 (Ω), for k ≥ 2. Finally, the following result gives us the finite speed of propagation property as seen in two different ways. The first statement describes how the compact support of solutions grows in time, while the second is a uniqueness type result. Note that these two statements are not equivalent since we are dealing with possible nonlinear terms. Both results will be used in the proof of the main result. For a proof of the theorem see [13]. Theorem 3.4. (Finite speed of propagation) Consider the problem (SW) under the hypothesis of Theorem 3.2. Then (1) if the initial data u0 , u1 is compactly supported inside the domain B(x0 , R) ∩ Ω, then u(x, t) = 0 outside B(x0 , R + t) ∩ Ω; (2) if (u0 , u1 ), (v0 , v1 ) are two pairs of initial data with compact support, with the corresponding solutions u(x, t), respectively v(x, t), with (u0 (x), u1 (x)) = (v0 (x), v1 (x)) for x ∈ B(x0 , R) ∩ Ω, then u(x, t) = v(x, t) inside B(x0 , R − t)∩Ω for any t < R. 4. Local existence of strong solutions This section contains our main result which will guarantee existence of H 2 solutions for the problem (SW) up to some time T > 0. The removal of the restriction (R) on the range of exponents p, m that control the growth of the source and damping terms is obtained by employing higher order energy estimates. Theorem 4.1 (Local existence in time of strong solutions). Let Ω ⊂ Rn be a bounded or unbounded domain in Rn with smooth C 2 boundary (possibly the entire Rn ), where 3 ≤ n < 6. Let f, g satisfy the assumptions (Af ) and (Ag ). Then there exists a time T > 0 such that the problem   utt − ∆u + f (u) + g(ut ) = 0, (x, t) ∈ Ω × [0, ∞) (SW) (u, ut )|t=0 = (u0 , u1 ), x ∈ Ω  u = 0, (x, t) ∈ ∂Ω × (0, T ). admits a strong solution on (0, T ). Remark 4.2. In case Ω = Rn the boundary condition for the above problem is removed. Proof. We start with a brief outline of the proof. First we consider the problem (SW) on a bounded domain Ω. We approximate the source term f with Lipschitz functions fε and obtain a sequence of approximating solutions uε . We show that for small initial data on these bounded domains, one has local existence in time of solutions. Next, we show that

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PETRONELA RADU

the smallness assumption is satisfied by appropriate truncations of our initial data on every small ball Bx0 (ρ) for every point x0 ∈ Ω, where the radius ρ is independent of x0 and ε. Thus we show that on every small ball the following bounds hold independently of ε: (4.1)

|∇uε |Bx0 (ρ) < C, |∇uεt |Bx0 (ρ) < C, |∆uε |Bx0 (ρ) < C, |uεtt |Bx0 (ρ) < C,

for all 0 < t < 1. We use these bounds and compactness arguments to pass to the limit in the source terms, and we use a monotonicity argument to pass to the limit in the damping terms. Thus, one obtains local existence of solutions on these small domains. Finally, we “patch” these small solutions to construct a solution on the whole domain Ω. The higher order estimates will yield the H 2 regularity of the solutions.

4.1. Compactness arguments. The goal of this section is to derive the bounds (4.1) which will allow us to pass to the limit in the sequence of approximations uε , and consequently, to obtain the local existence result on the bounded small domain Bx0 (ρ) (which is yet to be constructed). In a first step we assume that we are working with the following assumptions: (a1) Lipschitz source terms fε (the Lipschitz approximations fε are constructed exactly as in [13] and they satisfy the assumptions (Af ); more details will be provided later); (a2) on a bounded regular domain denoted by Ω; (a3) the initial data u0 , u1 are compactly supported inside a ball of radius R centered at a point x0 denoted by B(R) (since the equation (SW) is invariant by spatial translations one may assume w.l.o.g. that x0 is the origin). Thus, u0 has zero trace on ∂Ω, and hence so does the solution uε . (a4) the initial data has the improved regularity (u0 , u1 ) ∈ H 3 (Ω) × H 2 (Ω). According to Theorem 3.2 and Remark 3.3 this data will generate more regular solutions uε ∈ H 3 (Ω) ∩ H01 (Ω) and uεttt (·, t) ∈ L2 (Ω), for all t > 0. The additional regularity assumption is eliminated through a standard approximation argument (see [13] for details on this); (a5) “smallness” hypothesis of initial data: Z 1 1 2 2 (4.2) |∇u0 |Ω < λ, |u1 |Ω + |∇u0 |Ω + Fε (u0 (x)) dx < Φ1 (λ), 2 2 Ω where the function φ1 and the value λ are chosen in the sequel.

Next we will outline the “first order” potential well argument (as developed by Payne and Sattinger in [12, 15]) that gives bounds for |∇uε |Ω (we use the terminology of “first order” potential for obtaining bounds for the first order derivatives, to distinguish it from “second order” potential which estimates norms of second order derivatives), then we show that |∇uεt |Bx0 (ρ∗ ) < C for ρ∗ ≤ ρ sufficiently small. These bounds for the gradient will give us higher Lp regularity for uε and will also help us show the other two estimates from (4.1).


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4.1.1. First order estimates; “first order” potential well. We begin with the following remark. By the initial assumption (a3) and by part (1) of the finite speed of propagation property (Theorem 3.4) we have that u(t) ∈ H01 (B(R + t)). Hence, for t < 1 we have that u(t) ∈ H01 (Ω), where Ω := B(R + 1). (The upper bound for the time t < 1 is irrelevant since we will prove local existence in time.) 2n For q < n−2 and any v ∈ H01 (Ω), we have by Hölder’s inequality and Sobolev’s embedding theorem that the following inequality holds:

|v|q,Ω ≤ Cq (R + 1)

(4.3)

2n−q(n−2) 2q

|∇v|Ω ,

where (4.4)

Cq = C

∗

ω 2n−q(n−2) 2nq n

, n and the constant C ∗ is the (optimal) Sobolev’s constant that depends only on n and the volume of the unit sphere in Rn . Hence, the inequality (4.3) holds for uε (t), for all 0 < t < 1, where we choose q = p+1
2 0, q > 2. This is a C 1 function and it has one local maximum at x = (A0 q)−1/q−1 , whereas the above function φ1 may not be differentiable at the local maximum, or it may have two “humps”. Nevertheless, as we show in the sequel the arguments follow the same way for this slightly modified function. 1A

“classical” potential well is generated by a function Φ(x) =

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where by the constants A, B > 0 we denote: p+1 A = f1 Cp+1 (R + 1)

2n−(p+1)(n−2) 2

,

B = f2 C33 (R + 1)

6−n 2

.

We are interested in detecting the height of this generalized potential well generated by the function φ1 and where its maximum value is attained. Observe that for p > 1, Φ1 has at most three critical points on the positive semiaxis: x = 0 and possibly two local maxima, denoted by xc1 and xc2 . Let λ := min {xc1 , xc2 }. Then, we deduce that the first local maximum for Φ1 is attained at λ, by considering the following cases: x2 − Axp+1 . For 2 this function, the maximum is attained at xc1 = [(p + 1)A]−1/(p−1) . Hence, for the interval 1 [0, (B/A) p−2 ] the highest value will be attained by Φ1 at o n 1 x = max xc1 , (B/A) p−2 . 1

(i) If x < (B/A) p−2 , then max {Axp+1 , Bx3 } = Axp+1 ; hence, Φ1 (x) =

x2 − Bx3 , 2 1 whose maximum on [0, ∞) is attained at xnc2 = 1/(3B). On the interval [(B/A) p−2 , ∞) the o 1 maximum will be attained at at x = max xc2 , (B/A) p−2 . 1

(i) If x ≥ (B/A) p−2 then we have a similar analysis for the branch Φ1 (x) =

For our later analysis it is necessary to remark that the critical value λ depends on R, the radius of the ball which contains the support of the initial data. From (4.6) and the definition of Φ1 we obtain: Z 1 ε 2 |∇u (t)|Ω + Fε (uε (x, t)) dx ≥ Φ1 (|∇uε (t)|Ω ). 2 Ω Combining the above inquality, along with (4.5) and assumption (a5) we deduce: 1 ε 2 (4.8) |u (t)|Ω + Φ1 (|∇uε (t)|Ω ) 2 t Z 1 1 ε 2 ε 2 ≤ |ut (t)|Ω + |∇u (t)|Ω + Fε (uε (x, t)) dx 2 2 Ω Z 1 1 ≤ |u1 |2Ω + |∇u0 |2Ω + Fε (u0 (x)) dx < Φ1 (λ), 2 2 Ω from which we extract the bound (4.9)

Φ1 (|∇uε (t)|Ω ) < Φ1 (λ).

We have that t → |∇uε (t)|Ω is a continuous mapping; hence, should one assume that |∇uε (t)|Ω ≥ λ for some times t < 1 one would get Φ1 (|∇uε (t)|Ω ) ≥ Φ1 (λ) for those times t, due to the fact that |∇u0 |Ω < λ (in other words, we would start in the well with the initial data, but the solution would reach and surpass the maximum height, so it would leave the well). This comes in contradiction with (4.9), hence (4.10)

|∇uε (t)|Ω < λ for all t < 1.


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4.1.2. Second order estimates. By differentiating (SW), multiplying by uεtt , and integrating over Ω (see footnote 2) we obtain: Z uεtt uεttt − ∆uεt uεtt + fε0 (uε )uεt uεtt + g 0 (uεt )(uεtt )2 dx = 0 Ω

After a standard integration by parts we obtain: Z d ε 2 (utt ) + |∇uεt |2 + fε0 (uε )uεt uεtt + g 0 (uεt )(uεtt )2 dx = 0. Ω 2dt Since the damping term g is represented by an increasing differentiable function, we have that g 0 ≥ 0, so after integrating in time we obtain the following inequality: Z Z tZ Z ε 2 ε 2 0 ε ε ε (4.11) (utt ) + |∇ut | dx + 2 fε (u )ut utt dxds ≤ [utt (0)]2 + |∇u1 |2 dx Ω

0

Ω

Ω

We focus on the term involving the source fε0 (uε ) and estimate it by using Hölder’s inequality for 0 < t < 1 to obtain the following: Z tZ Z tZ d 0 ε ε ε 2 fε (u )ut utt dxds = fε0 (uε ) (uεt )2 dxds (4.12) dt 0 Ω 0 Ω Z Z tZ = fε0 (uε )(uεt )2 dx|s=t fε00 (uε )(uεt )3 dxds s=0 − Ω 0 Ω Z Z Z tZ ε p−1 ε 2 0 2 & − |u | (ut ) dx − |fε (u0 )|u1 dx − |uε |p−2 |uεt |3 dxds Ω

Ω

Z &− Ω

Z −

0

Ω

n2 Z n−2 Z n n(p−1) 2n ε ε |u | 2 dx |ut | n−2 dx − |fε0 (u0 )|u21 dx

|uε |

Ω 2n(p−2) 6−n

6−n Z 2n dx

Ω

Ω 2n

|uεt | n−2 dx

3(n−2) 2n

Ω

Note that the Hölder exponents are valid (i.e. they are greater than 1). Indeed, n2 and 2n n 2n together with their conjugates n−2 , respectively, 6−n are all greater than 1, since 3(n−2) 3 ≤ n < 6. We further estimate the terms in the above inequality and since p < n+2 we have by (4.3) n−2 that Z 4−(p−1)(n−2) n(p−1) n(p−1) (4.13) |uε | 2 dx ≤ C n(p−1) (R + 1) 2(p−1) (|∇uε |Ω ) 2 , 2 ZΩ 6−n−(n−2)(p−2) 2n(p−2) 2n(p−2) 2(p−2) |uε | 6−n dx ≤ C 2n(p−2) (R + 1) (|∇uε |Ω ) 6−n , Ω

6−n

where the constants on the right hand sides are computed with (4.4). Due to the fact that the initial data was chosen small enough so that the solution lives inside the first order 2Note

that all these operations are legitimate due to our higher regularity assumption (iv): (u, ut ) ∈ (H 3 (Ω) ∩ H01 (Ω) × H 2 (Ω)

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potential well given by the function Φ1 for all times t < 1, the norms of the gradients above are bounded as given by inequality (4.10); hence, from (4.13) we have Z Z 2n(p−2) n(p−1) ε (4.14) |u | 2 dx < M1 , |uε | 6−n dx < M2 , Ω

Ω

where M1 , M2 are given by M1 :=C n(p−1) (R + 1)

4−(p−1)(n−2) 2(p−1)

λ

n(p−1) 2

2

M2 :=C 2n(p−2) (R + 1)

6−n−(n−2)(p−2) 2(p−2)

λ

2n(p−2) 6−n

.

6−n

From inequality (4.3) applied to uεt we also have n Z n−2 Z 2n ε 2 ∗ ε n−2 |∇ut | dx . (4.15) |ut | dx ≤ C Ω

Ω

We return to (4.11) and use the estimate (4.12) together with the bounds (4.14), (4.15) to obtain 3/2 Z Z Z ε 2 ε 2 ε 2 (4.16) |∇ut | dx − A2 |∇ut | dx − B2 |∇ut | dx Ω Ω Ω Z Z 0 2 |f (u0 )|u1 dx + [utt (0)]2 + |∇u1 |2 dx, ≤ Ω

Ω

where A2 , B2 > 0 Based on the above estimate we construct the second order potential well function (4.17)

Φ2 (x) = (1 − A2 )x2 − B2 x3 .

We state the following: Claim 1. The constant A2 > 0 can be chosen such that A2 < 1. Assuming that Claim 1 holds true, the estimate (4.16) implies that Φ2 (|∇uεt |Ω ) < C. Since Φ2 (y) → ∞ as y → ∞ we obtain the desired bound (4.18)

|∇uεt |Ω < C.

Proof. (Claim 1). The constant A2 is generated by Sobolev’s embedding constants and by n(p−1) the L 2 and L2(p−2) norms of u on the domain Ω. The Sobolev constant itself depends on the domain Ω, but since these domains will be chosen to be balls, the constant does not n(p−1) increase when the size of the domain decreases. The L 2 and L2(p−2) norms of u are bounded by |∇uε |Ω according to the Sobolev embedding theorem, so one could decrease both these norms (and, consequently, decrease A2 ) by decreasing the radius of the domain


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(this follows from the equi-integrability of the ∇uε on Ω). Note, also, that |∇uε |Ω → 0 if |Ω| → 0. Hence A2 → 0 as |Ω| → 0. In section 4.2 we will construct small domains Bx0 (ρ) where the smallness assumptions on the initial data are satisfied; smallness of the initial data implies smallness for |∇uε |Ω , so we will choose those domains small enough so that we also have A2 < 1. We next use the multipliers −∆uεt and uεtt to complete our list of a priori estimates. Multiply the equation by −∆uεt and integrate by parts to obtain: Z ∇uεtt · ∇uεt + ∆uε · ∆uεt + fε0 (uε )∇uε · ∇uεt − g(uεt )∆uεt dx = 0. Ω

Note that Z −

g(uεt )∆uεt dx

Z =

Ω

g(uεt )|∇uεt |2 dx ≥ 0,

Ω

hence, by Hölder’s inequality we obtain the following estimates: Z Z d ε 2 ε 2 |∇ut | + |∆u | dx ≤ C |uε |p−1 |∇uε ||∇uεt |dx dt Ω Ω 1/β 1/α Z Z ε β ε (p−1)α |∇u | dx · |u | |dx ≤C Ω

Ω

Z

2n

|∇uεt | n−2 γ dx

1/γ

Ω 2n 2n . Since p < n+2 we clearly have that (p − 1)n < n−2 , so with α = n, β = 2, γ = n−2 n−2 by Sobolev’s emebedding theorems and the bounds (4.10) and (4.18) we obtain from the above inequality that for all t < 1

|∆uε |Ω < C.

(4.19)

By differentiating (SW) with respect to time and using the multiplier uεtt we obtain: Z Z Z d 0 ε ε ε ε 2 ε 2 |utt | + |∇ut | dx ≤ −2 fε (u )ut utt dx − 2 g 0 (uεt )(uεtt )2 dx. dt Ω Ω Ω After integrating in time from 0 to t < 1 and using the positivity of g 0 we obtain: Z Z ε 2 |utt | dx ≤ −2 fε0 (uε )uεt uεtt dxds Ω

Ω×(0,1)

By using the estimate (4.12) together with the bounds (4.10), (4.18) we obtain (4.20)

|uεtt |Ω < C.

14

PETRONELA RADU

4.2. Generating the “small” solutions. In this section we will show how to find the domains Bx0 (ρ) and how to construct on these domains pairs of initial data such that the assumptions (a2)-(a5) made in section 4.1 are satisfied. We start by finding a domain Bx0 (ρ) small enough, and construct a new pair of initial data (ux0 0 , ux1 0 ) such that (ux0 0 , ux1 0 ) satisfy the smallness assumptions (4.2) on Bx0 (ρ) and ux0 0 vanishes on the boundary of the ball. The balls Bx0 (ρ) have the radius ρ < 1 independent of x0 and small enough such that λ , 2

Φ(λ) , 8 p nλ Φ(λ) o 1 2(C ∗ ωn ) n |∇u0 |B(x0 ,ρ) + |u0 |B(x0 ,ρ) ≤ min , and 2 8 n+4 Φ(λ) 1 Φ(λ) 2 4C ∗ ωn2n |u0 |2B(x0 ,ρ) ≤ , |u1 |2B(x0 ,ρ) ≤ , 8 2 4 p+1 Φ(λ) f1 2 (C ∗ )p+1 |∇u0 |B(x0 ,ρ) + |u0 |B(x0 ,ρ) + 1 , ≤ p+1 ρ 8 3 Φ(λ) f2 ∗ 3 2 (C ) |∇u0 |B(x0 ,ρ) + |u0 |B(x0 ,ρ) + 1 ≤ , 3 ρ 8 where ωn is the volume of the unit ball in Rn and C ∗ is the constant from the Sobolev inequality. It can be easily shown that the above inequalities are satisfied by ρ such that ( r Φ1 (λ) λ λ , , (4.22) |∇u0 |B(x0 ,ρ) < min 1 , 2 8 4(C ∗ ωn ) n 1/(p+1) 1/3 ) (p + 1)Φ1 (λ) 3Φ1 (λ) 1 1 , , 2C ∗ 8m1 2C ∗ 8f2 (4.21)

|∇u0 |B(x0 ,ρ)
0. The positivity of γ amounts to p < which is guaranteed by assumption n−2 (Af ). Now we go back in (4.41) and take lim lim in both sides. First, in (4.42) take lim and δ→0 ε→0

δ→0

notice that we can bound the integrand the same way as in (4.43), and since we have the


19

convergence of the sets (0, t) × Bx0 (ρ) × (0, ρ/2)\A → (0, t) × Bx0 (ρ) × (0, ρ/2) as δ → 0, by Lebesgue dominated convergence theorem we have Z Z ε η lim lim fε (u )ut dxds = f (u)uηt dxds. δ→0 ε→0

(0,t)×Bx0 (ρ)\A

(0,t)×Bx0 (ρ)

From (4.43) we have that Z lim lim

δ→0 ε→0

1

|fε (uε )uηt |dxds ≤ lim C|A| γ lim M = 0,

A

δ→0

ε→0

where M is a bound for the first two factors on the RHS of (4.43). Thus we obtain: Z tZ Z tZ ε η lim fε (u )ut dxds = f (u)uηt dxds. ε→0

0

Bx0 (ρ)

0

Bx0 (ρ)

We R t R let η → 0 and by using (4.33) and (4.30) we obtain that the limit of this mixed term is f (u)ut dxds. The analysis of the second mixed term is similar and so we omit it. 0 Bx0 (ρ) Thus we conclude that (4.39) holds, hence (4.38) is valid. Note that (4.37), (4.38),(4.39) also gives us the continuity with respect to time u ∈ C(0, T ; H01 (Ω)) ∩ C 1 (0, T ; L2 (Ω)). The bounds (4.18), (4.19), (4.20) guarantee that our weak solution has the regularity of a strong solution, i.e. (u(·, t), ut (·, t)) ∈ (H 2 (Ω) ∩ H01 )(Ω) × H 1 (Ω) for all t > 0. 4.4. The patching argument. We conclude the proof of existence of solutions in this section, where we construct from small solutions the solution on the entire domain Ω.

4.4.1. Approximations on small domains. Let d > 0. Consider a lattice of points xk , k ∈ N in Ω situated at distance d away from each other, such that in every ball of radius d we find at least one xk . With ρ that satisfies (4.21) (where ρ depends only on the norms of the initial data), construct the balls of radius ρ centered at xk . The procedure outlined in subsection 4.2.1 for truncating the initial data around x0 to obtain a “small piece” denoted by (ux0 0 , u1x0 ), will be used now to construct around each xk the truncations (ux0 k , ux1 k ) which will satisfy the smallness assumptions |∇ux0 k |Bxk (ρ) < α, Z 1 xk 2 1 xk 2 + |∇u0 |Bx (ρ) + F (ux0 k (x)) dx < Φ(α). |u | k 2 1 Bxk (ρ) 2 Bxk (ρ) On each of the balls Bxk (ρ) we apply Theorem 3.2 to obtain global existence of solutions uε,xk for the problem (SW) with initial data (ux0 k , ux1 k ) and with the Lipschitz approximations fε for the source term.

20

PETRONELA RADU

4.4.2. Convergence of approximations on small domains. At this point, the arguments from section 4.3 for passing to the limit as ε → 0 in the sequence of approximate problems are applied, where x0 is successively replaced by different xk ’s. First, we apply Sattinger’s argument to estimate |∇uε,xk |2 on Bxk (ρ) for each k. (Note that we need to make use of the smallness assumptions written in Step 4.1.) The convergence uε,xk → uxk takes place on every domain Bxk (ρ) × (0, ρ/2), so we obtain a global (in time) solution to the boundary value problem (SW) for Bxk (ρ), for every k.

4.4.3. Patching small domains. The solutions uxk found in the above subsection 4.4.2 will now be “patched” together to obtain our general solution. First, we introduce the following notation. For k ∈ N let Ck := {(y, s) ∈ R3 × [0, ∞); |y − xk | ≤ ρ/2 − s} be the backward cones which have their vertices at (xk , ρ/2). For d small enough (i.e. for 0 < d < ρ/2) any two neighboring cones Ck and Cj will intersect and let Ik,j := Ck ∩ Cj . The maximum value for the time contained in Ik,j is equal to (ρ−d)/2 (see Figure 3 below). For t < ρ/2 we define the piecewise function: (4.44)

u(x, t) := uxk (x, t), if (x, t) ∈ Ck .

This solution is defined only up to time (ρ − d)/2, since this is the height of the intersection set of two cones with their vertices situated at distance d away from each other. By letting d → 0 we can obtain a solution that is well defined up to time ρ/2. Thus, we have u defined up to time ρ/2, which is the height of all cones Ck . Every pair (x, t) ∈ Rn × (0, ρ/2) belongs to at least one Ck , so in order to show that this function from (4.44) is well defined, we need to check that it is single-valued on the intersection of two cones. Also, we need to show that the above function is the solution generated by the pair of initial data (u0 , u1 ). Both proofs will be done in the next step.

4.4.4. Validity of solutions. In order to prove the properties that the previously constructed solution is well defined and satisfies the equation, we will go back and look at the solutions uxk as limits of the approximation solutions uε,xk . Consider first these small solutions on balls that do not intersect the boundary. For each k ∈ N we have (ux0 k , ux1 k ) = (u0 , u1 ) for every x ∈ Bxk (ρ) = {y ∈ Rn , |y − xk | < ρ/2} (see the construction of the truncations (ux0 k , ux1 k ) in subsection 4.2.1). Therefore, uε,xk (defined in section 4.4.1) is an approximation of the solution generated by the initial data (u0 , u1 ) on Ck (from the uniqueness property given by Part 2 of Proposition 3.4). We let ε → 0 (use


21

the argument from subsection 4.4.3) to show that the solution u on each Ck is generated by the initial data (u0 , u1 ). t 6

d ρ/2 (ρ − d)/2

@ Cj@

@ Ck@

@

@

@ @

@ @

Ik,j x k

ρ/2

@

@ @ @ xj -

@ @

-

x

Figure 3: The intersection of the domains Ck ∩ Ω and Cj ∩ Ω Remark 4.3. The above method of using cutoff functions and “patching” solutions based on uniqueness will work the same way in the case when the function f satisfies F (y) ≥ 0, y ∈ R and there is no bound on p. Since we can choose the height of the the cones arbitrarily large, the solutions exist globally in time under the positivity hypothesis for F . 5. Final remarks Remark 5.1. The paper shows that local well-posedness of strong solutions holds without assuming the restriction p 2n n n+2 (R) p+ < for 0, n−2 but for simplicity of the presentation we did not write the proof in its entire generality. References [1] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Editura Academiei, Bucuresti Romania and Noordhoff International Publishing, Leyden Netherlands (1976). [2] V. Barbu, I. Lasiecka, and M. A. Rammaha, On nonlinear wave equations with degenerate damping and source terms, Trans. Amer. Math. Soc., 357 (2005), 2571-2611. [3] L. Bociu Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping Nonlinear Analysis, 71, no. 12, (2009) 560–575. [4] L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete and Continuous Dynamical Systems A, 22, no. 4, (2008), 835–860. [5] L. Bociu and I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping Applicationes Mathematicae, [6] L. Bociu, P. Radu, Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping, (to appear in Proceedings of the 7th AIMS International Conference on Dynamical Systems, Differential Equations and Applications, Series B.) [7] L. C. Evans and R. F. Gariepy, “Measure Theory and Fine Properties of Functions,” Studies in Advanced Mathematics, (1992). [8] V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J.Differential Equations, 109, (1994), 295–308. [9] G. Lebeau, Non linear optic and supercritical wave equation. Hommage Pascal Laubin. Bull. Soc. Roy. Sci. Lige 70 (2001), no. 4-6, 267–306 (2002) [10] G. Lebeau, Perte de regularite pour les equations d’ondes sur-critiques. (French) [Loss of regularity for super-critical wave equations] Bull. Soc. Math. France 133 (2005), no. 1, 145–157. [11] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential and Integral Equations, vol.6, no.3, (1993), 507-533. [12] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math22 (1975), no. 3-4, 273-303. [13] P. Radu, Weak solutions to the Cauchy problem of a semilinear wave equation with damping and source terms, Advances in Differential Equations, 10, no. 11, (2005), 1261-1300 [14] P. Radu, Weak solutions to the initial boundary value problem for a semilinear wave equation, Applicationes Mathematicae, 35, no. 3, (2008), 355–378. [15] D. H. Sattinger, On global solutions of nonlinear hyperbolic equations, Arch. Rat. Mech. Anal., 30 (1968), 148-172. [16] J. Serrin, G. Todorova, and E. Vitillaro, Existence for a nonlinear wave equation with damping and source terms, Differential Integral Equations 16 (2003), no. 1, 13-50. [17] L. Tartar, Existence globale pour un système hyperbolique semi linéaire de la théorie cinétique des gaz, Séminaire Goulaouic-Schwartz (1975/1976), Equations aux dérivées partielles et analyse ´ fonctionnelle, Exp. No. 1, 11 pp. Centre Math., Ecole Polytech., Palaiseau, 1976. [18] L. Tartar, “ Topics in nonlinear analysis”, Publications Mathématiques d’Orsay, 78, 13, Université de Paris-Sud, Département de Mathématiques, Orsay, 1978, ii+271 pp.


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[19] G. Todorova, E. Vitillaro, Blow-up for nonlinear dissipative wave equations in Rn . J. Math. Anal. Appl. 303 (2005), no. 1, 242–257. [20] G. Todorova, B. Yordanov, On the regularizing effect of nonlinear damping in hyperbolic equations (preprint)

Department of Mathematics University of Nebraska-Lincoln 203 Avery Hall Lincoln, NE 68588-0130 USA e-mail:

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