04/06/2018 Fujita versus Strauss - a never ending story

Ore 15, Aula B3 (Palazzo Manfredini)

A lot of papers are devoted to the critical exponent $p_{crit}(n)$ in Cauchy problems for semilinear wave models with power-nonlinearity.

The model we have in mind is ($(t,x)\in [0,\infty)\times \mathbb{R}^n$)

\[ u_{tt} - \Delta u + b u_t + m^2 u=|u|^p,\,\,\,u(0,x)=\varphi(x),\,\,\,u_t(0,x)=\psi(x). \]

Here $b$ and $m^2$ are nonnegative constants. Critical exponent means that for some range of $p \geq p_{crit}(n)$ we have the global (in time) existence of small data Sobolev solutions.

On the contrary, for $1<p \leq p_{crit}(n)$ we have blow-up for Sobolev solutions under special assumptions for the data. \\

If $b=m^2=0$, then $p_{crit}(n)=p_0(n)$ is the well-known {\it Strauss exponent}. If $b=1$ and $m^2=0$, then $p_{crit}(n)=p_{Fuj}(n)$ is the well-known {\it Fujita exponent}.

In the talk we discuss a special semilinear wave model with scale-invariant time-dependent mass and dissipation and power-nonlinearity. We show how a competition between the Fujita exponent and Strauss exponent comes into play.

For a family of models we propose a new critical exponent.