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Instantaneous shrinking and localization of functions in \(Y_\lambda(m,p,q,N)\) and their applications. (English) Zbl 0992.35008
Summary: The aim of this paper is to discuss the instantaneous shrinking and localization of the support of functions in \(Y_\lambda (m,p,q,N)\) and their applications to some nonlinear parabolic equations including the porous medium equation \(u_t=\Delta u^m-u^q\), \(m>0\), \(q>0\) and the \(p\)-Laplace equation \(u_t=\text{div} (|\nabla u|^{p-2} \nabla u)-u^q\), \(p>1\), \(q>0\). In particular, as an application of the results, a necessary and sufficient condition for the solutions of the above \(p\)-Laplace equation with nonnegative finite Borel measures as initial conditions to have the instantaneous shrinking property of the support is obtained. This is an answer to an open problem posed by R. Kersner and A. Shishkov.
MSC:
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K55 Nonlinear parabolic equations
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