On the boundary blow-up solutions of p(x)-Laplacian equations with singular coefficient

Zhang, Qihu; Liu, Xiaopin; Qiu, Zhimei

doi:10.1016/j.na.2008.08.014

Cited by 10 publications

(4 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For an overview we refer to the surveys [14,21,30,36] and the monograph [13]. Apart from interesting theoretical considerations, the motivation to study such function spaces comes from applications to fluid dynamics [1,2,34], image processing [11], PDE and the calculus of variation [3,16,18,20,28,35,47].…”

Section: Introductionmentioning

confidence: 99%

Besov spaces with variable smoothness and integrability

Almeida

Hästö

2010

Journal of Functional Analysis

175

248

View full text Add to dashboard Cite

In this article we introduce Besov spaces with variable smoothness and integrability indices. We prove independence of the choice of basis functions, as well as several other basic properties. We also give Sobolev-type embeddings, and show that our scale contains variable order Hölder-Zygmund spaces as special cases. We provide an alternative characterization of the Besov space using approximations by analytic functions.

show abstract

Section: Introductionmentioning

confidence: 99%

Besov spaces with variable smoothness and integrability

Almeida

Hästö

2010

Journal of Functional Analysis

175

248

View full text Add to dashboard Cite

show abstract

“…Therefore, combining (29) and (31) we deduce the claim. On the other hand, from Definition 4, we obtain…”

Section: Proof We Consider the Test Functionmentioning

confidence: 66%

“…To the best of our knowledge, there are few results concerning the study of qualitative properties for parabolic equations with variable exponents by using this method. Furthermore, we shall also extend to the parabolic case some of the results by Zhang et al in [31], where radial sub-and supersolutions for some elliptic problems with variable exponents are constructed, and some of the results by Chung and Park in [22] and by Yuan et al in [27], to variable exponents case. In fact, we shall exploit their arguments in our parabolic problem setting with less conditions on the exponents ( , ), ( , ), and ( ) and the coefficient ( , ).…”

Section: Introductionmentioning

confidence: 79%

Qualitative Properties of Nonnegative Solutions for a Doubly Nonlinear Problem with Variable Exponents

Chaouai

Hachimi

2018

Abstract and Applied Analysis

View full text Add to dashboard Cite

We consider the Dirichlet initial boundary value problem ∂tum(x)-div⁡∇upx,t-2∇u=ax,tuq(x,t), where the exponents p(x,t)>1, q(x,t)>0, and m(x)>0 are given functions. We assume that a(x,t) is a bounded function. The aim of this paper is to deal with some qualitative properties of the solutions. Firstly, we prove that if ess⁡sup⁡p(x,t)-1<ess⁡inf⁡m(x), then any weak solution will be extinct in finite time when the initial data is small enough. Otherwise, when ess⁡sup⁡m(x)<ess⁡inf⁡p(x,t)-1, we get the positivity of solutions for large t. In the second part, we investigate the property of propagation from the initial data. For this purpose, we give a precise estimation of the support of the solution under the conditions that ess⁡sup⁡m(x)<ess⁡inf⁡p(x,t)-1 and either q(x,t)=m(x) or a(x,t)≤0 a.e. Finally, we give a uniform localization of the support of solutions for all t>0, in the case where a(x,t)<a1<0 a.e. and ess⁡sup⁡qx,t<ess⁡inf⁡p(x,t)-1.

show abstract

“…For more results for large solution of p-Laplace equations, we refer to [29][30][31][32] and the references therein. Our objective in this paper is to establish boundary asymptotic estimates for solutions of problem (1.1) under appropriate conditions on weight function b(x), which imposed a growth near ∂Ω, rather than requiring it to have a precise asymptotic behavior (see Remark 1.1), the nonlinear term f is a Γ -varying function, whose variation at infinity is not regular.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%