Slowly decaying solutions for semilinear wave equations in odd space dimensions

In this paper, we find the critical exponent for the existence of global small data solutions to: in the case of so-called non-effective damping, $\theta \in (\sigma ,2\sigma ]$, where $\sigma \ne 1$ and $f=|u{|}^{\alpha }$ or $f=|u_t{|}^{\alpha }$, in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers $\alpha >\tilde{\alpha }$ and do not exist, in general, for subcritical powers $1<\alpha <\tilde{\alpha }$. Assuming initial data to be small in $L^1$ or in some other $L^p$ space, $p\in (1,2)$, in addition to the energy space, the critical exponent only depends on the ratio $n/\left(\sigma p\right)$. We also prove the global existence of small data solutions in high space dimension for $\alpha >\bar{\alpha }$, but we leave open to determine if a counterpart nonexistence result for $\alpha <\bar{\alpha }$ holds or not.

We study the Cauchy problem for the Euler-Poisson-Darboux equation, with a power nonlinearity: where $\mu >0$, $p>1$ and $\alpha >-2$. Here either $t_0=0$ (singular problem) or $t_0>0$ (regular problem). We show that this model may be interpreted as a semilinear wave equation with borderline dissipation: the existence of global small data solutions depends not only on the power p, but also on the parameter μ. Global small data weak solutions exist if In the case of $\alpha =0$, the above condition is equivalent to $p>p_{\mathrm{crit}}=\max \{p_{\mathrm{Str}}(1+\mu ),3\}$, where $p_{\mathrm{Str}}(k)$ is the critical exponent conjectured by W.A. Strauss for the semilinear wave equation without dissipation (i.e. $\mu =0$) in space dimension k. Varying the parameter μ, there is a continuous transition from $p_{\mathrm{crit}}=\infty$ (for $\mu =0$) to $p_{\mathrm{crit}}=3$ (for $\mu \ge 4/3$). The optimality of $p_{\mathrm{crit}}$ follows by known nonexistence counterpart results for $1<p\le p_{\mathrm{crit}}$ (and for any $p>1$ if $\mu =0$).

As a corollary of our result, we obtain analogous results for generalized semilinear Tricomi equations and other models related to the Euler-Poisson-Darboux equation.

In this note we study the global existence of small data solutions to the Cauchy problem for the semilinear wave equation with a not effective scale-invariant damping term, namely where $p>1$, $n\ge 2$. We prove blow-up in finite time in the subcritical range $p\in (1,p_2(n)]$ and existence theorems for $p>p_2(n)$, $n=2,3$. In this way we find the critical exponent for small data solutions to this problem. Our results lead to the conjecture $p_2(n)=p_0(n+2)$ for $n\ge 2$, where $p_0(n)$ is the Strauss exponent for the classical semilinear wave equation with power nonlinearity.