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A Cauchy problem for \(u_ t-\Delta u=u^ p\) with \(0<p<1\). Asymptotic behaviour of solutions. (English) Zbl 0601.35051

Existence, uniqueness and regularity of global solutions for the Cauchy problem \[ u_ t-\Delta u=u^ p\quad in\quad (0,\infty)\times {\mathbb{R}}^ n;\quad u(0,x)=u_ 0(x)\geq 0\quad in\quad {\mathbb{R}}^ n;\quad u\geq 0\quad in\quad (0,\infty)\times {\mathbb{R}}^ n \] is proved for a large class of non identically null initial data. Solutions are shown to be uniformly above \(((1-p)t)^{1/(1-p)}\) which is the maximal solution of the above problem with initial data \(u_ 0=0\). Finally the existence of self-similar solutions (not necessarily radial) describing the asymptotic behaviour of solutions as t goes to infinity is shown.

MSC:

35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35A30 Geometric theory, characteristics, transformations in context of PDEs
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References:

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