Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations

• Published in: Journal of Differential Equations Vol. 228; no. 2; pp. 611 - 632
• Main Authors: Chill, Ralph, Fiorenza, Alberto
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.09.2006
• Subjects: Convergence of solutions; Decay rate; Orlicz–Sobolev space; Quasilinear parabolic problems; Łojasiewicz–Simon inequality; secondary; Decay rate; Łojasiewicz–Simon inequality; Orlicz–Sobolev space; Convergence of solutions; Quasilinear parabolic problems; primary
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2006.02.009

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation u t − div a ( x , ∇ u ) + f ( x , u ) = 0 on a bounded domain, subject to Dirichlet boundary and to initial conditions. The data are supposed to satisfy suitable regularity and growth conditions. Our approach to the convergence result and decay estimate is based on the Łojasiewicz–Simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates. The abstract results and their applications are discussed also in the framework of Orlicz–Sobolev spaces.