Global infinite energy solutions of the critical semilinear wave equation
The author considers the Cauchy problem in \(\mathbb R^d\) for the semilinear wave equation \[ (\partial^2_t-\Delta)u +| u| ^{p-1}=0, \quad u_{t=0}=u_0 \quad\text{and}\quad \partial_t u_{t=0}=u_1 \] with critical exponent \(p=\frac{2d}{2-d} -1\). Using the method of Bourgain he proves that there exists a global solution if the initial data are of the form \((u_o,u_1)= (v_o+w_0, v_1+w_1)\) with \(E=\| v_o| \dot{H}_1\| + \| v_1| L_2 \| <\infty\) and \(\| w_o| \dot{B}^1_{2,\infty}\| + \| v_1| \dot{B}^0_{2,\infty} \| < C \exp(-\exp E^\kappa)\), \(\kappa,C >0\), \(d=3,4,6\). Moreover it is proved that there exists a global solution if the initial data are of the form \((u_o,u_1)= (v_o+c_0/| x| ^2, v_1+ c_1/| x| ^3)\) with \(\| v_o| \dot{H}_1\| + \| v_1| L_2 \| <\infty\), \(\max(c_0,c_0)<\varepsilon\) and \(d=6\). The last statement is proved by the Calderón method.
scientific article; zbMATH DE number 5361870