A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films

Abstract

<p style='text-indent:20px;'>In this paper, the initial-boundary value problem for a class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films is studied. By means of the theory of potential wells, the global existence, asymptotic behavior and finite time blow-up of weak solutions are obtained.</p>

A Class of Fractional Viscoelastic Kirchhoff Equations Involving Two Nonlinear Source Terms of Different Signs

“…Liu et al [30] introduced a family of potential wells that were different from those in Refs. [27][28][29]. In contrast, the definition of the potential well in this paper differs from those in the above studies.…”

“…[25] to deal with a fourth-order damped nonlinear hyperbolic equation. Xu and Su [27], Luo et al [28], and Liu and Li [29] introduced a family of potential wells to study the equation under consideration, respectively. Liu et al [30] introduced a family of potential wells that were different from those in Refs.…”

Initial Boundary Value Problem for a Fractional Viscoelastic Equation of the Kirchhoff Type

“…The potential well was first proposed by Sattinger [27] in order to study the global existence of solutions to a nonlinear hyperbolic equation. Subsequently, it was widely employed to analyze the qualitative properties of the solutions to evolution equations (see, for example, [18,[28][29][30][31][32][33][34][35][36][37][38][39] and the references therein), and it has now developed into a theoretical system.…”

Note on a higher order pseudo‐parabolic equation with variable exponents

“…Liu and Li in [9] considered the $$$ p $$$‐Laplacian fourth‐order parabolic equation: $$$$ {u}_t-\Delta u+{\Delta}^2u-\operatorname{div}\left({\left|\nabla u\right|}^{p-2}\nabla u\right)=f(u),\kern0.30em \left(x,t\right)\in \Omega \times \left(0,T\right). $$$$ They obtained the sufficient conditions on the global existence, asymptotic behavior, and finite‐time blow‐up of weak solutions.…”

“…Inspired by the works [9][10][11]20], we want to study the singularity of weak solutions to problem (1.1). To our best knowledge, there were rare results for weak solutions to (1.1).…”