On L^r-regularity of global attractors generated by strong solutions of reaction-diffusion equations

In this paper we prove that the global attractor generated by strong solutions of a reaction-diffusion equation without uniqueness of the Cauchy problem is bounded in suitable L^r-spaces. In order to obtain this result we prove first that the concepts of weak and mild solutions are equivalent under an appropriate assumption.

Also, when the nonlinear term of the equation satisfies a supercritical growth condition the existence of a weak attractor is established.