The Role of Critical Exponents in Blowup Theorems

Date
1990
Authors
Levine, Howard
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Altmetrics
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Research Projects
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Mathematics
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Abstract

In this article various extensions of an old result of Fujita are considered for the initial value problem for the reaction-diffusion equation $u_t = \Delta u + u^p $ in $R^N $ with $p > 1$ and nonnegative initial values. Fujita showed that if $1 < p < 1 + {2 / N}$, then the initial value problem had no nontrivial global solutions while if $p > 1 + {2 / N}$, there were nontrivial global solutions. This paper discusses similar results for other geometries and other equations including a nonlinear wave equation and a nonlinear Schrödinger equation.

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This is an article from SIAM Review 32 (1990): 262, doi:10.1137/1032046. Posted with permission.

Keywords
critical exponents, parabolic, hyperbolic equations, Schrödinger equations, global existence, global nonexistence
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