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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:
 35K65 Parabolic equations of degenerate type 35K15 Second order parabolic equations, initial value problems 35K55 Nonlinear parabolic equations 35B45 A priori estimates for solutions of PDE
##### Keywords:
$p$-Laplacian equation; porous media equation
Full Text:
##### References:
 [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. [2] Ladyzenskaja, O. A.: New equations for the description of incompressible fluids and solvability in the largest boundary value problem for them. Proc. Steklov inst. Math. 102, 95 (1976) [3] Andreucci, D.; Dibenedetto, E.: On the Cauchy problem and initial traces for a class of evolution equations with strongly nonlinear sources. Ann. scuola norm. Sup. Pisa cl. Sci. (4) 18, 363-441 (1991) · Zbl 0762.35052 [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 [5] Ladyzenskaja, O. A.; Solonnikov, V. A.; Ural’ceve, N. N.: Linear and quasilinear equations of parabolic type. (1968) [6] Giaquinta, M.: Multiple integrals in the calculus of variations and nonlinear elliptic systems. (1983) · Zbl 0516.49003 [7] Ivanov, A. V.: Hölder estimates for quasilinear doubly degenerate parabolic equations. J. sov. Math. 56, 2330-2347 (1991) · Zbl 0729.35018 [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