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Title of the item:

Uniform Bounds for Solutions to Quasilinear Parabolic Equations

Title :
Uniform Bounds for Solutions to Quasilinear Parabolic Equations
Authors :
Cipriani, Fabio
Grillo, Gabriele
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Source :
Journal of Differential Equations; November 2001, Vol. 177 Issue: 1 p209-234, 26p
We consider a class of quasilinear parabolic equations whose model is the heat equation corresponding to the p-Laplacian operator, u=Δpu≔∑di=1∂i(|∇u|p−2∂iu) with p∈[2, d), on a domain D⊂Rdof finite measure. We prove that |u(t, x)|⩽c|D|αt−β‖u0‖γrfor all t>0,x∈Dand for all initial data u0∈Lr(D), provided ris not smaller than a suitable r0, where α, β, γare positive constants explicitly computed in terms of d, p, r. The nonlinear cases associated with the case p=2 display exactly the same contractivity properties which hold for the linear heat equation. We also show that the nonlinear evolution considered is contractive on any Lqspace for any q∈[2, +∞].

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