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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.

##### MSC:
 35K65 Degenerate parabolic equations 35B35 Stability in context of PDEs 35K55 Nonlinear parabolic equations 35K15 Initial value problems for second-order parabolic equations
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##### References:
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