Dec 29, 1989 - Blow up for a diffusion-advection equation. Nicholas D. Alikakos. Department of Mathematics, University of Tennessee, Knoxville, TN 37996,.
Proceedings of the Royal Society of Edinburgh, 113A, 181-190, 1989
Blow up for a diffusion-advection equation Nicholas D. Alikakos Department of Mathematics, University of Tennessee, Knoxville, TN 37996, U.S.A. Peter W. Bates Applied Mathematics, National Science Foundation, Washington, D.C. 20550, U.S.A. and Christopher P. Grant* Department of Mathematics, Brigham Young University, Provo, UT 84602, U.S.A. (MS received 29 November 1988. Revised MS received 16 May 1989)
Synopsis These results describe the asymptotic behaviour of solutions to a certain non-linear diffusionadvection equation on the unit interval. The "no flux" boundary conditions prescribed result in mass being conserved by solutions and the existence of a mass-parametrised family of equilibria. A natural question is whether or not solutions stabilise to equilibria and if not, whether they blow up in finite time. Here it is shown that for non-linearities which characterise "fast association" there is a critical mass such that initial data which have supercritical mass must lead to blow up in finite time. It is also shown that there exist initial data with arbitrarily small mass which also lead to blow up in finite time.
1. Introduction This article deals with the initial-boundary value problem
ux+f(u) = 0, u(x, 0) = uo(x),
x = 0, 1, 0 Mc and fix time r =
__Ji_^!_____.
(3.1}
We claim that the solution, u, to (2.1) with initial data u0 blows up by time T. Suppose this is not the case, then u exists and is bounded beyond t = T as noted near the beginning of Section 2. Define z: [0, 1] x [0, T]^> [0, M] by (3 2)
'
where a = (q - 2)1 (q - 1). We will show that z is a subsolution for problem (2.3). For convenience, let us name the two triangular parts of the domain of z as Aj = {(JC, 0 e [0, 1] x [0, T\. 0 ^ x ^ 1 - t/T},
A2={(x,t)e[0, l]x[0,
T]:l-t/TMc, Lemma 4.3 shows that wy blows up at y = 0 at some time r^T. This contradicts our assumption and thereby proves the theorem. • This theorem by itself does not show that there actually exist initial data with subcritical mass which give rise to solutions which blow up. Theorem 1.2 will be established by exhibiting initial data with arbitrarily small mass which also satisfy the hypotheses of Theorem 4.5. Proof of Theorem 1.2. With A € (0, 1] and k > 0 to be determined later, define uo(x) =
(kk)a+1
(x + kf
with a = (q - 2)/(q - 1). Using this as initial data in (2.1), the corresponding initial condition in (2.3) is
Setting f(x) = v'o(x) + (vo(x))q, we seek to determine the minimum over [0,1]. Using the expression in (4.6) we obtain nX)
2(ar
3 (x + A) A)3
(x + A)2"
and a short computation shows that / has a unique critical point 1/(2,-3)
Blow up for a diffusion-advection equation
189
It is clear that for k large and A sufficiently small, jce[0, 1]. Since q > \, it is obvious that /(*)—»+°° as x—* — A+. It is equally obvious that f(x)—>0~ as x—»+°°; therefore, the unique critical point, x, must correspond to a negative minimum of/. One may calculate
and so condition (4.4) of Theorem 4.5 may be written
(kk\a + B(kk\-^^-^ 2 for some choice of M > Mc, where B is the factor independent of kk in f(x). Since a = 2 — q/(q — 1) one sees that (4.7) is independent of A and that for any choice of M> Mc, k may be chosen so large that (4.7) holds. Therefore by first fixing k sufficiently large, and then taking A > 0 arbitrarily close to zero, we find that the mass of « 0 , being vo(l), may be taken arbitrarily small, while still giving rise to a solution which blows up in finite time. • Acknowledgment We are happy to acknowledge here a number of stimulating discussions Fife from which we gained some useful insight. We should also acknowledge the support of the National Science Foundation, and the and Engineering Research Council of Great Britain through a grant to Watt University.
with P. like to Science Heriot-
References 1 H. Amann. Parabolic evolution equations and non-linear boundary conditions. J. Differential Equations 72 (1988), 201-269. 2 P. Benilan. Lecture notes (University of Kentucky at Lexington, 1981). 3 M. Bertsch, M. Gurtin, D. Hilhorst and L. Peletier. On interacting populations that disperse to avoid crowding; The effect of sedentary colony. J. Math. Biol. 19 (1984), 1-12. 4 M. G. Crandall and L. Tartar. Some relations between non-expansive and order preserving mappings. Proc. Amer. Math. Soc. 78 (1980), 385-390. 5 H. Fujita. Foundations of Ultracentrifugal Analysis (Chichester: John Wiley and Sons, 1975). 6 B. H. Gilding. Improved theory for a non-linear degenerate parabolic equation. Ann. Scuola Norm. Sup. Pisa Cl. Set. (4) (to appear). 7 C. P. Grant. Masters Thesis (Brigham Young University, 1988). 8 Y. Giga and R. Kohn. Characterizing blow up using similarly variables. Indiana Univ. Math. J. 36 (1987), 1-40. 9 M. E. Gurtin and A. C. Pipkin. A note on interacting populations that disperse to avoid crowding. Quart. Appl. Math. 42 (1984), 87-94. 10 D. Henry. Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (Berlin: Springer, 1981). 11 O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva. Linear and Quasilinear Equations of Parabolic Type (Providence, R.I.: American Mathematical Society, 1968). 12 H. Levine, L. Payne, P. Sacks and B. Straughan. (II). Analysis of a convective reaction-diffusion equation. SIAM J. Math. Anal. 20 (1989) 133-147. 13 T. Nagai. Some non-linear degenerate diffusion equations with a non-locally convective term in ecology. Hiroshima Math. J. 13 (1983), 449-464. 14 F. Saidi. The Large Time Behaviour of Solutions of a Diffusion Equation Involving a Non-local Convective Term (Ph.D. Thesis, University of Maryland, Baltimore County, 1988).
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15 N. Shigesada, K. Kawasaki and E. Teramoto. Spatial segregation of interacting species. J. Theoret. Biol. 79 (1979), 89-99. 16 A. Yoshikawa. On the behaviour of the solutions to the Lamm equation of the ultracentrifuge. SIAMJ. Math. Anal. 15 (1984), 686-711. 17 T. I. Zelenyak. Stabilisation of solutions of boundary value problems for a second-order equation with one space variable. Differential Equations 4 (1968), 17-22. (Issued 29 December 1989)
