| Contents |
| In this talk we study the influence of a nonlinear memory
\[ F (t, u) =\int_0^t (t-s)^{-\gamma} |u(s, x)|^p ds,\qquad \gamma\in(0,1),\]
with $p>1$, on the global existence of small data solutions to
\[ \begin{cases}
u_{tt}-\triangle u + \mu u_t = F(t,u),\qquad t\geq0, \quad x\in\mathbb{R}^n, \\
u(0,x)=u_0(x),\\
u_t(0,x)=u_1(x),
\end{cases} \]
where $\mu>0$, in space dimension $1\leq n\leq 5$.
We prove the global existence for $p> \overline{p}(n,\gamma)$, where $\overline{p}(n,\gamma)$ is the critical exponent, i.e. no global weak solution exists for $1 < p \leq \overline{p}(n,\gamma)$ for suitable, arbitrarily small, data in some Sobolev space.
We also show how the critical exponent changes if we consider a problem with structural damping
\[ \begin{cases}
u_{tt}-\triangle u + \mu(-\triangle)^{\frac12} u_t = F(t,u),\qquad t\geq0, \quad x\in\mathbb{R}^n, \\
u(0,x)=u_0(x),\\
u_t(0,x)=u_1(x),
\end{cases} \]
and how the stronger dissipation allow us to obtain a global existence result in any space dimension $n\geq2$.
References
[1] M. D'Abbicco, The influence of a nonlinear memory on the damped wave equation, Nonlinear Analysis 95 (2014), 130--145, doi:10.1016/j.na.2013.09.006.
[2] M. D'Abbicco, A wave equation with structural damping and nonlinear memory, Nonlinear Differential Equations and Applications, to appear, doi:10.1007/s00030-014-0265-2. |
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