Regularisation effects of nonlinear semigroups

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One introduces natural and simple methods to deduce Lˢ-L∞-regularisation estimates for 1≤ s< ∞ of nonlinear semigroups holding uniformly for all time with sharp exponents from natural Gagliardo-Nirenberg inequalities. From Lq-Lʳ Gagliardo-Nirenberg inequalities, 1≤ q, r≤ ∞, one deduces Lq-Lʳ estimates for the semigroup. New nonlinear interpolation techniques of independent interest are introduced in order to extrapolate such estimates to Lq-L∞ estimates for some q, 1≤ q<∞. Finally one is able to extrapolate to Lˢ-L∞ estimates for 1≤ 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.