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Search: MSC category 35K57 ( Reaction-diffusion equations )

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1. CJM Online first

Phan, Tuoc
Lorentz estimates for weak solutions of quasi-linear parabolic equations with singular divergence-free drifts
This paper investigates regularity in Lorentz spaces of weak solutions of a class of divergence form quasi-linear parabolic equations with singular divergence-free drifts. In this class of equations, the principal terms are vector field functions which are measurable in $(x,t)$-variable, and nonlinearly dependent on both unknown solutions and their gradients. Interior, local boundary, and global regularity estimates in Lorentz spaces for gradients of weak solutions are established assuming that the solutions are in BMO space, the John Nirenberg space. The results are even new when the drifts are identically zero because they do not require solutions to be bounded as in the available literature. In the linear setting, the results of the paper also improve the standard Calderón-Zygmund regularity theory to the critical borderline case. When the principal term in the equation does not depend on the solution as its variable, our results recover and sharpen known, available results. The approach is based on the perturbation technique introduced by Caffarelli and Peral together with a "double-scaling parameter" technique, and the maximal function free approach introduced by Acerbi and Mingione.

Keywords:gradient estimate, quasi-linear parabolic equation, divergence-free drift
Categories:35B45, 35K57, 35K59, 35K61

2. CJM 2009 (vol 62 pp. 74)

Ducrot, Arnaud; Liu, Zhihua; Magal, Pierre
Projectors on the Generalized Eigenspaces for Neutral Functional Differential Equations in $L^{p}$ Spaces
We present the explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues for linear neutral functional differential equations (NFDE) in $L^{p}$ spaces by using integrated semigroup theory. The analysis is based on the main result established elsewhere by the authors and results by Magal and Ruan on non-densely defined Cauchy problem. We formulate the NFDE as a non-densely defined Cauchy problem and obtain some spectral properties from which we then derive explicit formulas for the projectors on the generalized eigenspaces associated with some eigenvalues. Such explicit formulas are important in studying bifurcations in some semi-linear problems.

Keywords:neutral functional differential equations, semi-linear problem, integrated semigroup, spectrum, projectors
Categories:34K05, 35K57, 47A56, 47H20

3. CJM 1997 (vol 49 pp. 798)

Yu, Minqi; Lian, Xiting
Boundedness of solutions of parabolic equations with anisotropic growth conditions
In this paper, we consider the parabolic equation with anisotropic growth conditions, and obtain some criteria on boundedness of solutions, which generalize the corresponding results for the isotropic case.

Keywords:Parabolic equation, anisotropic growth conditions, generalized, solution, boundness
Categories:35K57, 35K99.

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