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Article Dans Une Revue Dynamics of Partial Differential Equations Année : 2019

Perturbations of self-similar solutions

Résumé

We consider the nonlinear heat equation $u_t = \Delta u + |u|^\alpha u$ with $\alpha >0$, either on ${\mathbb R}^N $, $N\ge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2) \alpha <4$, for every $\mu \in {\mathbb R}$, if the initial value $u_0$ satisfies $u_0 (x) = \mu |x-x_0|^{-\frac {2} {\alpha }}$ in a neighborhood of some $x_0\in \Omega $ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)= u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $\mu |x-x_0|^{-\frac {2} {\alpha }}$ on ${\mathbb R}^N $. Moreover, if $\mu \ge \mu _0$ for a certain $ \mu _0( N, \alpha )\ge 0$, and $u_0 I\ge 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)= u_0$, but there are infinitely many sign-changing solutions.

Dates et versions

hal-01826926 , version 1 (30-06-2018)

Identifiants

Citer

Thierry Cazenave, Flavio Dickstein, Ivan Naumkin, Fred B. Weissler. Perturbations of self-similar solutions. Dynamics of Partial Differential Equations, 2019, 16 (2), pp.151-183. ⟨10.4310/DPDE.2019.v16.n2.a3⟩. ⟨hal-01826926⟩
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