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Register a new userHarnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coeffcients from Kato-type classes
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For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.
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For a general class of divergence type quasi-linear degenerate parabolic equations with differentiable structure and lower order coefficients form bounded with respect to the Laplacian we obtain $L^q$-estimates for the gradients of solutions, and for the lower order coefficients from a Kato-type class we show that the solutions are Lipschitz continuous with respect to the space variable.
We continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form begin{eqnarray} text{div}big(A(x,u, abla u)big) = B(x,u, abla u)text{ for }xinOmega onumber end{eqnarray} as considered in our previous paper giving local boundedness of weak solutions. Here we derive a version of Harnacks inequality as well as local Holder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin and N. Trudinger for quasilinear equations, as well as ones for subelliptic linear equations obtained by Sawyer and Wheeden in their 2006 AMS memoir article.
We prove matrix and scalar differential Harnack inequalities for linear parabolic equations on Riemannian and Kahler manifolds.
For a class of singular divergence type quasi-linear parabolic equations with a Radon measure on the right hand side we derive pointwise estimates for solutions via the nonlinear Wolff potentials.
We give a proof of Holder continuity for bounded local weak solutions to the equation $u_t= sum_{i=1}^N (|u_{x_i}|^{p_i-2} u_{x_i})_{x_i}$, in $Omega times [0,T]$, with $Omega subset subset mathbb{R}^N$, under the condition $ 2<p_i<bar{p}(1+2/N)$ for each $i=1,..,N$, being $bar{p}$ the harmonic mean of the $p_i$s, via recently discovered intrinsic Harnack estimates. Moreover we establish equivalent forms of these Harnack estimates within the proper intrinsic geometry.
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