Attractors for parabolic equations with nonlinear boundary conditions, critical exponents, and singular initial data

Abstract

The title of the paper says it all. The author considers a reaction-diffusion equation with nonlinear boundary conditions on a bounded domain Ω⊂R N . The initial value u 0 is allowed to be a function in L r (Ω) , with 1<r<∞ , as well as a measure on Ω . The nonlinear reaction term f and the nonlinearity g that is part of the Neumann-type boundary condition are both assumed to be locally Lipschitz and of critical growth at infinity. The author refers to his previous works, joint with Arrieta, Carvalho, Oliva, Pereira and Tadjine, for background results leading to this work. The key assumption in the present study is a balance term between the nonlinearities f and g , (3.1), (5.1) or (6.5) in the paper, which the author interprets as "a condition that reflects a competition among diffusion, reaction and boundary flux''. In the case of power-like nonlinearities, the author obtains sharper results. Now you can read the title of the paper again.