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.