Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem

Lee, Tzong-Yow; Ni, Wei-Ming

doi:10.1090/S0002-9947-1992-1057781-6

Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem
HTML articles powered by AMS MathViewer

by Tzong-Yow Lee and Wei-Ming Ni PDF

Trans. Amer. Math. Soc. 333 (1992), 365-378 Request permission

Abstract:

We investigate the behavior of the solution $u(x,t)$ of \[ \left \{ {\begin {array}{*{20}{c}} {\frac {{\partial u}} {{\partial t}} = \Delta u + {u^p}} \hfill & {{\text {in}}\;{\mathbb {R}^n} \times (0,T),} \hfill \\ {u(x,0) = \varphi (x)} \hfill & {{\text {in}}\;{\mathbb {R}^n},} \hfill \\ \end {array} } \right .\] where $\Delta = \sum \nolimits _{i = 1}^n {{\partial ^2}/\partial _{{x_i}}^2}$ is the Laplace operator, $p > 1$ is a constant, $T > 0$, and $\varphi$ is a nonnegative bounded continuous function in ${\mathbb {R}^n}$. The main results are for the case when the initial value $\varphi$ has polynomial decay near $x = \infty$. Assuming $\varphi \sim \lambda {(1 + |x|)^{ - a}}$ with $\lambda$, $a > 0$, various questions of global (in time) existence and nonexistence, large time behavior or life span of the solution $u(x,t)$ are answered in terms of simple conditions on $\lambda$, $a$, $p$ and the space dimension $n$.

References

D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76. MR 511740, DOI 10.1016/0001-8708(78)90130-5
Avner Friedman, Partial differential equations of parabolic type, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR 0181836
Hiroshi Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t}=\Delta u+u^{1+\alpha }$, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124 (1966). MR 214914
Alain Haraux and Fred B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982), no. 2, 167–189. MR 648169, DOI 10.1512/iumj.1982.31.31016
Kantaro Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad. 49 (1973), 503–505. MR 338569
Stanley Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963), 305–330. MR 160044, DOI 10.1002/cpa.3160160307
Kusuo Kobayashi, Tunekiti Sirao, and Hiroshi Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977), no. 3, 407–424. MR 450783, DOI 10.2969/jmsj/02930407
Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), no. 2, 262–288. MR 1056055, DOI 10.1137/1032046
David H. Sattinger, Topics in stability and bifurcation theory, Lecture Notes in Mathematics, Vol. 309, Springer-Verlag, Berlin-New York, 1973. MR 0463624
Xuefeng Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc. 337 (1993), no. 2, 549–590. MR 1153016, DOI 10.1090/S0002-9947-1993-1153016-5
Fred B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), no. 1-2, 29–40. MR 599472, DOI 10.1007/BF02761845