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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.

35K65Parabolic equations of degenerate type
35K15Second order parabolic equations, initial value problems
35K55Nonlinear parabolic equations
35B45A priori estimates for solutions of PDE
Full Text: DOI
[1] A. S. Kalashnikov, Some problems of the qualitative theory of nonlinear degenerate second-order parabolic equations, Uspekhi Mat. Nauk421987, 135--176 (in Russian); English transl. Russian Math. Surveys421987, 169--222.
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[4] Zhao, J.: On the Cauchy problem and initial traces for the evolution of p-Laplacian equations with strongly nonlinear sources. J. differential equations 121, 329-383 (1995) · Zbl 0836.35081
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[8] Zhao, J.; Xu, Z.: Cauchy problem and initial traces for a doubly nonlinear degenerate parabolic equation. Sci. China (Ser. A) 39, 673-684 (1996) · Zbl 0860.35068