It is shown that if $u$ solves the wave equation in $\mathbb{R}^{4+1}$ with initial data $u(\cdot,0) = u_0(\cdot) \in H^s$ and $\partial_tu(\cdot,0) = u_1(\cdot ) \in H^{s-1}$, where $0.5<s \leq 0.55$, then $u(x,t) \to u_0(x)$ and $\partial_tu(x,t) \to u_1(x)$ pointwise as $t \to 0$, for all $x$ outside an exceptional set of Hausdorff dimension at most $6-4s$. In a very small range of $s$, this verifies a conjecture of Barcel\'o, Bennett, Carbery, and Rogers. More generally, a partial improvement to the exceptional set bound in $\mathbb{R}^{n+1}$ is obtained for $n \geq 4$ and $1/2<s<n/4$.
Let $\mathbb H^n$ be the $n$-dimensional hyperbolic space, and $\Delta_{\mathbb H^n}$ be the Laplace--Beltrami operator on $\mathbb H^n$. Let $u=u(t,\cdot)$ be the solution to the wave equation $\partial^2_{t}u-\big(\Delta_{\mathbb H^n}+\frac{1}{4}(n-1)^2\operatorname{Id}\big)u=0$ with initial conditions $u(0,x)=f(x)$...
Peng Chen, X. Duong, Yun-Xiang Wang et al.· 1 citation· ⚡1
In this paper, we study the Cauchy problem $u_t-\mathcal L u=h(t)u^p$, $u(0)=\varepsilon u_0\ge0$, where $p>1$, $h\ge0$ is only locally integrable, and $\mathcal L=\sum_{j=1}^mX_j^2$ is a homogeneous H\"ormander sum of squares on $\mathbb R^n$ of homogeneous dimension $q$. Diffusion spreads the solution out and weakens...
We study the maximal existence time $T^{*}(\varrho)$ of the solution of the semilinear wave equation $u_{tt}-\Delta u=u|u|^{p-1}-u_{t}|u_{t}|^{q-1}$ in a bounded domain, with Dirichlet boundary condition and initial data $(\varrho f,\varrho g)$, where $1<q<p$ and the amplitude $\varrho$ is large. For nontrivial $f$ and...
Let $G$ be the group $\mathbb{R}_+\ltimes \mathbb{R}^n$ endowed with Riemannian symmetric space metric $d$ and the right Haar measure $\mathrm{d} \rho$ which is of $ax+b$ type, and $L$ be the positive definite distinguished left-invariant Laplacian on $G$. Let $u=u(t,\cdot)$ be the solution to $u_{tt}+Lu=0$ with initia...
We consider the Dirichlet problem $-\Delta u=|x|^\alpha u^p$, $u>0$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega\subset R^N$, $N\ge 3$, is a bounded domain containing the origin and $\alpha>0$. We prove that a solution exists for every exponent $1<p<p_\alpha=\frac{N+2+2\alpha}{N-2}$, in particular in the whole...
We study the lifespan \(T^*(\varepsilon)\) of classical solutions to the Cauchy problem for the one-dimensional quasilinear wave equation $u_{tt}-(c(u)^2u_x)_x=0$, with initial data of size $\varepsilon>0$. We assume that $c'(\theta)\sim a|\theta|^{p-2}$ as $\theta \to 0$, where \(a>0\) and \(p>3\) or $p=2$. We obtain...
Yuusuke Sugiyama, Taro Yamanoi· 0 citations
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