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Article Dans Une Revue American Journal of Mathematics Année : 2020

Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value

Résumé

We consider the nonlinear heat equation $u_t - \Delta u = |u|^\alpha u$ on ${\mathbb R}^N$, where $\alpha >0$ and $N\ge 1$. We prove that in the range $0 < \alpha <\frac {4} {N-2}$, for every $\mu >0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= \mu |x|^{-\frac {2} {\alpha }}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.

Dates et versions

hal-01556681 , version 1 (05-07-2017)

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Citer

Thierry Cazenave, Flavio Dickstein, Ivan Naumkin, Fred B. Weissler. Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value. American Journal of Mathematics, 2020, 142 (5), pp.1439-1495. ⟨10.1353/ajm.2020.0037⟩. ⟨hal-01556681⟩
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