Convergence of Anisotropically Decaying Solutions of a Supercritical Semilinear Heat Equation |
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Authors: | Peter Polá?ik Eiji Yanagida |
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Institution: | (1) School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA;(2) Mathematical Institute, Tohoku University, Sendai 980-8578, Japan |
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Abstract: | We consider the Cauchy problem for a semilinear heat equation with a supercritical power nonlinearity. It is known that the
asymptotic behavior of solutions in time is determined by the decay rate of their initial values in space. In particular,
if an initial value decays like a radial steady state, then the corresponding solution converges to that steady state. In
this paper we consider solutions whose initial values decay in an anisotropic way. We show that each such solution converges
to a steady state which is explicitly determined by an average formula. For a proof, we first consider the linearized equation
around a singular steady state, and find a self-similar solution with a specific asymptotic behavior. Then we construct suitable
comparison functions by using the self-similar solution, and apply our previous results on global stability and quasi-convergence
of solutions. |
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Keywords: | Semilinear parabolic equation Critical exponent Anisotropic decay Quasi-convergence Self-similar solution |
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