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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 $\\tilde{q}$, $1\\le \\tilde{q}<\\infty$. 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