Global weighted estimates for quasilinear elliptic equations with non-standard growth

In this paper we obtain a new global gradient estimates in weighted Lorentz spaces for weak solutions of p(x)$p(x)$-Laplacian type equation with small BMO coefficients in a δ-Reifenberg flat domain. The modified Vitali covering lemma, the maximal function technique and the appropriate localization method are the main analytical tools. Our results improve the known results for such equations.     

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.     

Weighted Lorentz estimates for parabolic equations with non-standard growth on rough domains

The main aim of this paper is to prove the Calderón–Zygmund estimates for a general nonlinear parabolic equation of p(x, t)-Laplacian type in the weighted Lorentz spaces. Note that we only require some mild conditions on the nonlinearity of coefficients and the underlying domain. The result for these nonlinear parabolic equations is new even in the particular case when the growth p(x, t) is a constant.     

Global weighted estimates for second-order nondivergence elliptic and parabolic equations

In this paper, we obtain the global weighted \(L^p\) estimates for second-order nondivergence elliptic and parabolic equations with small BMO coefficients in the whole space. As a corollary, we obtain \(L^p\)-type regularity estimates for such equations.     

Global Lorentz estimates for nonlinear parabolic equations on nonsmooth domains

Consider the nonlinear parabolic equation in the form

$$\begin{aligned} u_t-\mathrm{div}{\mathbf {a}}(D u,x,t)=\mathrm{div}\,(|F|^{p-2}F) \quad \text {in} \quad \Omega \times (0,T), \end{aligned}$$

where \(T>0\) and \(\Omega \) is a Reifenberg domain. We suppose that the nonlinearity \({\mathbf {a}}(\xi ,x,t)\) has a small BMO norm with respect to x and is merely measurable and bounded with respect to the time variable t. In this paper, we prove the global Calderón-Zygmund estimates for the weak solution to this parabolic problem in the setting of Lorentz spaces which includes the estimates in Lebesgue spaces. Our global Calderón-Zygmund estimates extend certain previous results to equations with less regularity assumptions on the nonlinearity \({\mathbf {a}}(\xi ,x,t)\) and to more general setting of Lorentz spaces.

     

Global weighted estimates for the gradient of solutions to nonlinear elliptic equations

We consider nonlinear elliptic equations of p  -Laplacian type that are not necessarily of variation form when the nonlinearity is allowed to be discontinuous and the boundary of the domain can go beyond the Lipschitz category. Under smallness in the BMO nonlinearity and sufficient flatness of the Reifenberg domain, we obtain the global weighted L^q$L^q$ estimates with q∈(p,∞)$q\in (p,\infty )$ for the gradient of weak solutions.     

Gradient estimates for doubly nonlinear parabolic equations

Local gradient estimates for weak solutions of the equation

Local estimates for parabolic equations with nonlinear gradient terms

In this paper we deal with local estimates for parabolic problems in ${\mathbb{R}^N}$ with absorbing first order terms, whose model is $$\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.$$ where ${T >0 , \, N\geq 2,\, 1 < q \leq 2,\, f(t,x)\in L^1\left( 0,T; L^1_{\rm loc} \left(\mathbb{R}^N\right)\right)}$ and ${u_0\in L^1_{\rm loc}\left(\mathbb{R}^{N}\right)}$ .     

Gradient estimates for parabolic equations in generalized weighted Morrey spaces

We consider the Cauchy–Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg flat domain.The coefficients supposed to be only measurable in one of the space variables and small BMO with respect to the others.We obtain Calderón–Zygmund type estimate for the gradient of the solution in generalized weighted Morrey spaces with Muckenhoupt weight.     

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.     

Error estimates for discontinuous Galerkin method for nonlinear parabolic equations

We consider the nonlinear parabolic partial differential equations. We construct a discontinuous Galerkin approximation using a penalty term and obtain an optimal L^∞(L²) error estimate.     

Uniform blow-up rate for parabolic equations with a weighted nonlocal nonlinear source

In this paper, the blow-up rate of solutions of semi-linear reaction-diffusion equations with a more complicated source term, which is a product of nonlocal (or localized) source and weight function a(x), is investigated. It is proved that the solutions have global blow-up, and that the rates of blow-up are uniform in all compact subsets of the domain. Furthermore, the blow-up rate of _∞|u(t)| is precisely determined.     

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.      

Partial regularity for parabolic systems with non-standard growth

We study non-linear parabolic systems with non-standard p(z)-growth conditions and establish that the gradient of weak solutions is locally H?lder continuous with H?lder exponent b ? (0,1){\beta \in (0,1)} with respect to the parabolic metric on an open set of full Lebesgue measure, provided the exponent function p(z) itself is H?lder continuous with exponent β with respect to the parabolic metric.     

A priori estimates for solutions to systems of nonlinear parabolic equations

A class of nondiagonal systems of nonlinear parabolic equations that can be reduced to a scalar parabolic equation in the phase space of a larger dimension is described. In view of such a reduction, it is possible to state the maximum principle for solutions to systems of nonlinear parabolic equations and derive a priori C^2+α-estimates for a solution to the Cauchy problem. Bibliography: 19 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 41–67.     

Maximum modulus estimates for generalized solutions of doubly nonlinear parabolic equations

Maximum modulus estimates are obtained for generalized solutions of doubly nonlinear parabolic equations (DNPE). The equation

Global existence and decay estimates for solutions of degenerate parabolic equations

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionInthispaper,weconsiderthefollowinginitial--boundaryvalueproblemwhereQ~fix(o,co),aQ=aflx(o,co),fiisaboundeddomaininEuclideanspaceR"(n22)withsmoothboundaryonandac=(u.,,'Iu..)denotesthegradientoffunctionu(x).Weassumethefunctionsal(x,t,u,p)(i=1,2,',n)anda(x,t,u,p)arelocallyH5ldercontinuousonfix(0,co)suchthatwherealtuandparepositiveconstants,m,aZIa3.hi,b2,alIadZ20,or321areconstants,m*E[0,m 2),hi16z/0,afl m*/…     

一类非线性抛物型方程组解的整体存在及爆破

研究了一类带有非线性边界条件的非线性抛物型方程组解的整体存在及解在有限时刻爆破问题.通过构造方程组的上、下解.得到了解整体存在及解在有限时刻爆破的充分条件.对指数型反应项和边界流采用了常微分方程方法构造其上下解,而其它例如第一特征值等方法运用于该方程就比较困难.