Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Authors: Tzong-Yow Lee and Wei-Ming Ni
Journal: Trans. Amer. Math. Soc. 333 (1992), 365-378
MSC: Primary 35K55; Secondary 35B30, 35B40
MathSciNet review: 1057781
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Abstract: We investigate the behavior of the solution $ u(x,t)$ of

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\frac{{\partial u}} {{\partial ... ...hi (x)} \hfill & {{\text{in}}\;{\mathbb{R}^n},} \hfill \\ \end{array} } \right.$

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 + \vert x\vert)^{ - 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$.

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Keywords: Semilinear parabolic Cauchy problem, global existence, life span, large time asymptotic behaviors
Article copyright: © Copyright 1992 American Mathematical Society