Study of weak solutions for a fourth-order parabolic equation with variable exponent of nonlinearity

Abstract

The aim of this paper is to study the existence and uniqueness of weak solutions for an initial boundary problem of a fourth-order parabolic equation with variable exponent of nonlinearity. First, the authors of this paper apply Leray-Schauder's fixed point theorem to prove the existence of solutions of the corresponding nonlinear elliptic problem and then obtain the existence of weak solutions of nonlinear parabolic problem by combining the results of the elliptic problem with Rothe's method. In addition, the authors also discuss the regularity of weak solutions in the case of space dimension one. (2000). 35K35 · 35K65 · 35B40. Mathematics Subject Classification

Study of weak solutions to a nonlinear fourth‐order parabolic equation with boundary degeneracy

“…The existence and regularity of weak solutions were obtained therein, and Liang and Zheng 14 gave the existence and asymptotic behavior of solutions for a nonlinear source problem. Besides, the existence of the corresponding variable exponent problem was considered by Guo and Gao (see their work 15 ) and Lianget al (see their work 16 ). If the mobility function has a double degenerate construction, Liang 17 has obtained the existence of positive steady‐state solutions with homogeneous conditions.…”

Energy classification in a nonstandard fourth‐order parabolic equation with a Navier boundary condition

“…For more results on parabolic problems with nonstandard growth conditions, we refer the interested readers to the works. [19][20][21][22][23][24][25][26] And the recently related parabolic equations in the frame of potential well, we refer the interested readers to the works. [27][28][29] Inspired by the works, 1,16,18 we will study the singular solutions to problem (1.1).…”

“…It is interesting that the blow‐up rates are determined by the maximal of the variable exponents. For more results on parabolic problems with nonstandard growth conditions, we refer the interested readers to the works 19–26 . And the recently related parabolic equations in the frame of potential well, we refer the interested readers to the works 27–29 …”

A degenerate operator in non divergence form

“…In [12] the authors consider a degenerate fourth order operator of the form ∇ • (m(u)∇∆u) where m is a specific function, proving existence and non uniqueness results for the parabolic equation associated to this operator (see also [4], [5], [7], [14], [20] or [25]). In [21], the existence of a weak solution for the following equation is proved…”