Regularisation effects of nonlinear semigroups

Abstract

One introduces natural and simple methods to deduce $L^s$-$L^{\infty }$-re\-gularisation estimates for $1\le s<\infty$ of nonlinear semigroups holding uniformly for all time with sharp exponents from natural Gagliardo-Nirenberg inequalities. From $L^q$-$L^r$ Gagliardo-Nirenberg inequalities, $1\le q,r\le \infty$, one deduces $L^q$-$L^r$ estimates for the semigroup. New nonlinear interpolation techniques of independent interest are introduced in order to extrapolate such estimates to $L^{\tilde{q}}$-$L^{\infty }$ estimates for some $\tilde{q}$, $1\le \tilde{q}<\infty$. Finally one is able to extrapolate to $L^s$-$L^{\infty }$ estimates for $1\le s<q$. The theory developed in this monograph allows to work with minimal regularity assumptions on solutions of nonlinear parabolic boundary value problems as illustrated in a plethora of examples including nonlocal diffusion processes.