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On the Cauchy problem and initial traces for a degenerate parabolic equation. (English) Zbl 0691.35047
We consider the Cauchy problem \[ u_ t-div(| Du|^{p- 2}Du)=0\quad in\quad R^ N\times (0,\infty),\quad p>2;\quad u(x,0)=u_ 0(x),\quad x\in R^ N \] and discuss existence of solutions in some strip \(S_ T=R^ N\times (0,T)\), \(0<T\leq \infty\), in terms of the behaviour of \(u_ 0=u_ 0(x)\) as \(| x| \to \infty\). The results obtained are optimal in the class of nonnegative locally bounded solutions, for which a Harnack-type inequality holds. Uniqueness is shown under the assumption that the initial values are taken in the sense of \(L^ 1_{loc}(R^ N)\). Our discussion generalizes the already investigated case \(N=1\) and the equation \[ u_ t-\Delta | u|^{m-1}u=0,\quad m>0 \] of porous medium, but our methods enlargen results obtained for these cases.
Reviewer: M.Krüger

MSC:
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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