Local gradient estimates for nonlinear elliptic equations

In this paper we obtain local Lq, q?p, gradient estimates for weak solutions of elliptic equations of p-Laplacian type with small BMO coefficients.     

Self-similar solutions of quasilinear parabolic equations with nonlinear gradient terms

This paper is devoted to study the classification of self-similar solutions to the m ≥ 1,p,q ＞ 0 and p + q ＞ m. For m = 1, it is shown that the very singular self-similar solution exists if and only if nq + (n + 1)p ＜ n + 2, and in case of existence, such solution is unique. For m ＞ 1, it is shown that very singular self-similar solutions exist if and only if 1 ＜ m ＜ 2 and nq + (n + 1)p ＜ 2 + mn, and such solutions have compact support if they exist. Moreover, the interface relation is obtained.     

Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data $C^1$ up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known as the generalized Dirichlet problem. Therefore, our result implies loss of boundary conditions in finite time. Specifically, a solution satisfying homogeneous boundary conditions in the viscosity sense eventually becomes strictly positive at some point of the boundary.     

Pointwise gradient estimates of solutions to onedimensional nonlinear parabolic equations

We present here an improved version of the method introduced by the first author to derive pointwise gradient estimates for the solutions of one-dimensional parabolic problems. After considering a general qualinear equation in divergence form we apply the method to the case of a nonlinear diffusion-convection equation. The conclusions are stated first for classical solutions and then for generalized and mild solutions. In the case of unbounded initial datum we obtain several regularizing effects for t > 0. Some unilateral pointwise gradient estimates are also obtained. The case of the Dirichlet problem is also considered. Finally, we collect, in the last section, several comments showing the connections among these estimates and the study of the free boundaries associated to the solutions of the diffusion-convection equation.     

Global estimates for nonlinear parabolic equations

We consider nonlinear parabolic equations of the type $$u_t - {\rm div}a(x, t, Du)= f(x, t) \quad {\rm on}\quad \Omega_T =\Omega\times (-T,0),$$ under standard growth conditions on a, with f only assumed to be integrable. We prove general decay estimates up to the boundary for level sets of the solutions u and the gradient Du which imply very general estimates in Lebesgue and Lorentz spaces. Assuming only that the involved domains satisfy a mild exterior capacity density condition, we provide global regularity results.     

Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|^q−1u|∇u|^p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞.     

Controllability for parabolic equations with uniformly bounded nonlinear terms

We consider a control system for a parabolic equation in a Banach space with uniformly bounded nonlinear termF,

Gradient estimates for doubly nonlinear parabolic equations

Local gradient estimates for weak solutions of the equation

Some gradient estimates and Harnack inequalities for nonlinear parabolic equations on Riemannian manifolds

In this paper, by the method of J. F. Li and X. J. Xu (Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456–4491 ), we shall consider the nonlinear parabolic equation on Riemannian manifolds with , . First of all, we shall derive the corresponding Li–Xu type gradient estimates of the positive solutions for . As applications, we deduce Liouville type theorem and Harnack inequality for some special cases. Besides, when , our results are different from Li and Yau's results. We also extend the results of J. F. Li and X. J. Xu, and the results of Y. Yang.     

Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data

We obtain existence results for some strongly nonlinear Cauchy problems posed in and having merely locally integrable data. The equations we deal with have as principal part a bounded, coercive and pseudomonotone operator of Leray-Lions type acting on , they contain absorbing zero order terms and possibly include first order terms with natural growth. For any p > 1 and under optimal growth conditions on the zero order terms, we derive suitable local a-priori estimates and consequent global existence results.     

Complement of gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds

In this paper, along the idea of Souplet and Zhang, we deduce a local elliptic‐type gradient estimates for positive solutions of the nonlinear parabolic equation: on for α ≥ 1 and α ≤ 0. As applications, related Liouville‐type theorem is exported. Our results are complement of known results. Copyright © 2016 John Wiley & Sons, Ltd.     

Lorentz estimates for asymptotically regular fully nonlinear parabolic equations

We prove a global Lorentz estimate of the Hessian of strong solutions to the Cauchy–Dirichlet problem for a class of fully nonlinear parabolic equations with asymptotically regular nonlinearity over a bounded C^{1, 1} domain. Here, we mainly assume that the associated regular nonlinearity satisfies uniformly parabolicity and the ‐vanishing condition, and the approach of constructing a regular problem by an appropriate transformation is employed.     

Global gradient estimates for degenerate parabolic equations in nonsmooth domains

This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for the gradient. The results extend to the parabolic systems as well. The higher integrability estimates provide a useful tool in several applications.      

Local gradient estimates of solutions to some conformally invariant fully nonlinear equations

We establish local gradient estimates to solutions of general conformally invariant fully nonlinear second order elliptic equations. To cite this article: Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 343 (2006).