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Cauchy problem and initial trace for a doubly degenerate parabolic equation with strongly nonlinear sources. (English) Zbl 0993.35057
Let $S_T=\bbfR^N\times (0,T)$, $N\ge 1$ and $T>0$. The author investigates in $S_T$ the Cauchy problem for the equation $u_t=\operatorname {div}(|Du^m|^{p-2}Du^m)+u^q$ with the initial condition $u(x,0)=u_0(x)$. Here $p>1$, $m>0$, $m(p-1)>1$, $q>1$, and $u_0$ is locally integrable in $\bbfR^N$. Of course, for $m=1$ we have the familiar evolution $p$-Laplacian equation, and for $p=2$ we have the porous media equation. The Cauchy problem for the general case is investigated for a large class of initial conditions. To describe this class, the following norm is defined for $h\ge 1$. $|||f|||_h=\sup_{x\in \bbfR^N}\|f\|_h(B_1(x))$. Here, $\|.\|_h$ represents the usual norm in $L^h(B_1(x))$, where $B_1(x)$ denotes the unit ball centered at $x$ in $\bbfR^N$. The first result is the following. Assume $u_0\ge 0$, $|||u_0|||_h<\infty$, where $h=1$ if $q<m(p-1)+p/N$ and $h>(N/p)(q-m(p-1))$ otherwise. Then there is a constant $T_0>0$ depending on the data such that a solution $u(x,t)$ exists in $S_{T_0}$. Quantitative bounds for the solution and results involving a supersolution are obtained. Also the problem of uniqueness is discussed.

MSC:
35K65Parabolic equations of degenerate type
35K15Second order parabolic equations, initial value problems
35K55Nonlinear parabolic equations
35B45A priori estimates for solutions of PDE
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References:
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