The Cauchy problem for a semilinear wave equation. II

L. V. Kapitanskii

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 21, Tome 182 (1990), pp. 38-85

Résumé

The Cauchy problem for a semilinear hyperbolic equation of the form $$ \frac{\partial^2}{\partial t^2}u(t,x)+iB(t)\frac\partial{\partial t}u(t,x)+A(t)u(t,x)+f(t,x;u(t,x))=0\qquad{(1)} $$ is studied. In (1): $x$ runs through a smooth Riemannian manifold $\mathfrak{M}$ without boundary, $\mathrm{dim}\,\mathfrak{M}=n\geqslant3$, $A(t)$ and $B(t)$ are time-dependent pseudodifferential operators (on $\mathfrak{M}$) of order 2 and $\leqslant1$, resp., with real principal symbols $a_2(t,x,\xi)$ and $b_1(t,x,\xi)$, and $a_2(t,x,\xi)\geqslant\nu|\xi|^2$, $\nu>0$, all $t$ and $(x,\xi)\in T^*\mathfrak{M}\setminus0$. Under certain assumptions on the nonlinearity $f$ which, in the special case of differential operators $A(t)$ and $B(t)$ and $f(t,x;z)=\lambda|z|^{\rho-1}z$, reduce to $\lambda\geqslant0$ and $1\leqslant\rho\leqslant(n+2)/(n-2)$ (the critical value $\rho=(n+2)/(n-2)$ is allowed.), we prove that for arbitrary initial data $$ u(0)=\varphi\in H^1,\quad \frac\partial{\partial t}u(0)=\psi\in L_2\qquad{(2)} $$ there exists and is unique the global in $t$ weak solution $u$ of the problem (1), (2).