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On some parabolic equations involving superlinear singular gradient terms. (English) Zbl 1473.35351

Summary: In this paper we prove existence of nonnegative solutions to parabolic Cauchy-Dirichlet problems with (eventually) singular superlinear gradient terms. The model equation is \[ u_t-\Delta_pu=g(u)|\nabla u|^q+h(u)f(t,x)\quad\text{in }(0,T)\times\Omega, \] where \(\Omega\) is an open bounded subset of \(\mathbb{R}^N\) with \(N>2\), \(0<T<+\infty\), \(1<p<N\), and \(q<p\) is superlinear. The functions \(g,\,h\) are continuous and possibly satisfying \(g(0)=+\infty\) and/or \(h(0)=+\infty\), with different rates. Finally, \(f\) is nonnegative and it belongs to a suitable Lebesgue space. We investigate the relation among the superlinear threshold of \(q\), the regularity of the initial datum and the forcing term, and the decay rates of \(g,\,h\) at infinity.

MSC:

35K92 Quasilinear parabolic equations with \(p\)-Laplacian
35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K67 Singular parabolic equations
35R06 PDEs with measure
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References:

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