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Uniform bounds for solutions to quasilinear parabolic equations. (English) Zbl 1036.35043
The authors consider a class of quasilinear parabolic equations on a domain $$D \subset \mathbb{R}^d$$ of finite Lebesgue measure in the form $u_t(t,x) = \text{div\,} a(t,x,u(t,x), \nabla u(t,x)); \quad t \in (0,\infty),\;x \in D.$ where $$a : (0,\infty)\times D \times \mathbb{R} \times \mathbb{R}^d \to \mathbb{R}^d$$ is a Carathéodory function satisfying the conditions $a(t,x,u,\xi).\xi \geq C_1 | \xi| ^p,\qquad | a(t,x,u,\xi)| \leq C_2 | \xi| ^{p-1},$ almost everywhere for positive constants $$C_1$$, $$C_2$$, $$d \geq 3$$, $$2 \leq p \leq d$$. This class admits (among others) the $$p$$-Laplacian as a corresponding elliptic operator.
One of the main results of the paper is the global uniform ultracontractive bound $\| u(t)\| _{\infty}\leq C \frac{| D| ^{\alpha}}{t^{\beta}}\| u(0)\| ^{\gamma}_{q_0}$ valid for a suitable choice of $$\alpha, \beta, \gamma, q_0$$. Moreover, contractivity of the corresponding evolutionary process, i.e. the inequality $\| u(t,.)\| _q \leq \| u(0,.)\| _q$ for any $$t > 0, q \in [2, \infty)$$ is proved.
The fundamental step in the proof is a study of a function $y(s)= \log (\| u(s,.)\| _{r(s)}).$ For a chosen function $$r(s)$$ it is differentiable and satisfies a differential inequality, whose integration gives the required result. In deducing the differential inequality the authors use a new type of energy-entropy inequality similar to Gross logarithmic Sobolev inequalities.

##### MSC:
 35B45 A priori estimates in context of PDEs 35K15 Initial value problems for second-order parabolic equations 35K65 Degenerate parabolic equations 35K55 Nonlinear parabolic equations
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