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On the Cauchy problem for the evolution \(p\)-Laplacian equations with gradient term and source. (English) Zbl 1128.35059
The authors study local and global existence of solutions of the Cauchy problem for the equation \[ u_t=\text{div}\,(| Du| ^{p-2}Du)+| u| ^{q-1}u-\lambda| Du| ^l,\qquad x\in\mathbb R^N,\;t>0, \] where \(p>2\), \(q\geq1\), \(l\geq1\) and \(\lambda\in\mathbb R\). They prove local existence and uniqueness of solutions for suitable class of initial data provided \(l\leq p-1\) and \(l\geq p/2\), respectively. If \(p/2\leq l\leq p-1<q<p-1+p/N\) then positive solutions with large initial data blow up in finite time. On the other hand, if \(l\leq p-1\) and \(q>p-1+p/N\) then positive solutions with small initial data exist globally.

35K65 Degenerate parabolic equations
35B35 Stability in context of PDEs
35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
Full Text: DOI
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