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Global solutions for some nonlinear parabolic equations with nonmonotonic perturbations. (English) Zbl 0595.35058
The following problems are studied: \[ (E_ 1)\quad u_ t- \sum^{N}_{i=1}(| u_{x_ i}|^ m u_{x_ i})_{x_ i}+\beta (x,u)=0,\quad u(x,0)=u_ 0\in W_ 0^{1,m+2} \]
\[ and\quad E_ 2\quad u_ t-\Delta (| u|^ m u)+\beta (x,u)=0,\quad u(x,0)=u_ 0\in W_ 0^{1,m+2} \] in a bounded domain for the x- variable in \(R^ N\), with locally Hölder continuous \(\beta\) and \(| \beta (x,u)| \leq k_ 0 | u|^{\alpha +1}.\)
The author proves the global existence and a priori estimates of the solutions provided the norm of initial data is sufficiently small in \(L^{p+2}\) with p depending on N, m and \(\alpha\).
Reviewer: B.Nowak

MSC:
35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35B45 A priori estimates in context of PDEs
35B20 Perturbations in context of PDEs
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