Existence of a global solution to a semi-linear wave equation with initial data of non-compact support in low space dimensions
Consider the following initial-value problem: \[ u_ tt - \Delta u = A | u |^ p,\;(x,t) \in \mathbb{R}^ n \times (0,\infty), \quad u(x,0) = f(x),\;u_ t (x,0) = g(x),\;x \in \mathbb{R}^ n \tag{*} \] where \(n=2\) or \(n=3\), \(A\) and \(p\) are positive constants with \(p>1\), \(f \in C^ 3(\mathbb{R}^ n)\) and \(g \in C^ 2(\mathbb{R}^ n)\). In this field it is interesting to find a critical number \(p^*>1\) such that if \(p>p^*\) then we obtain a nontrivial global solution for some \(f\) and \(g\), and if \(1<p<p^*\) then we get a blowing-up (in finite time) solution. The author improves these assertions obtained so far in the following form: Theorem 1. Let \(n=2\). Assume \(p>p^* \equiv (3 + \sqrt {17})/2\) and \[ \sum_{| \alpha | \leq 3} | D^ \alpha_ xf | + \sum_{| \alpha | \leq 2} | D^ \alpha_ xg | \leq {\varepsilon \over (1+b \psi)^{1+k}}, \quad x \in \mathbb{R}^ 2, \tag{1} \] where \(K \geq 2/(p-1)\), \((p \neq 5)\), \(K>1/2\) \((p=5)\) and \(\varepsilon>0\) is small enough. Then there exists a unique global \(C^ 2\)-solution for \((*)\). Theorem 2. Let \(n=3\). Assume \(p>1+ \sqrt 2\) and (1) holds in \(\mathbb{R}^ 3\) with \(K \geq 2/(p-1)\) \((p \neq 3)\), \(K>1\) \((p=3)\). Then a similar assertion as in Theorem 1 holds.
scientific article; zbMATH DE number 482116