The Cauchy problem in abstract Gevrey spaces for a nonlinear weakly hyperbolic equation of second order
The authors investigate the existence of local solutions to the abstract Cauchy problem: \[ (1)\quad u''+ A(t)u= f(t, u(t)), \qquad\quad (2)\quad u(0)= u_ 0, \quad u'(0)= u_ 1, \] in Hilbert space \(H\), where \(A(t)\) is a nonnegative unbounded operator. They introduce the Hilbert triple \(V\subseteq H\subseteq V'\) in some way and assume that the operator \(A(t)\) satisfies (for \(0<\lambda\leq 2\), \(T>0)\), \(A(t)\in C^ \lambda ([0,T];{\mathcal L}(V,V'))\), what for \(0<\lambda \leq 1\) means Hölder continuity of exponent \(\lambda\), and for \(1<\lambda \leq 2\) means Hölder continuity of exponent \(\lambda -1\), of the first time derivative. Moreover they assume that the equation (1) is weakly hyperbolic, i.e. for all \(v,w\in V\), \(\langle Av,v \rangle\geq 0\), \(\langle Av,w\rangle= \overline{\langle Aw,v \rangle}\) as function \(f\) satisfies some hypotheses. In the paper the abstract Gevrey spaces \(X_ r^ s\) of order \(s\geq 0\) and the space \(X_{0+}^ s= \bigcup_{r>0} X_ r^ s\) are introduced. The main result states that for all \(u_ 0, u_ 1\in X_{0+}^ s\), there exists \(T_ 0>0\) such that problem (1) (2) has a unique solution in \(C^ 1 ([0,T_ 0]; X_{0+}^ s)\), provided \(1\leq s\leq 1+ \lambda/2\), \(s<2\). The proof is based on the method of infinite order energy.
scientific article; zbMATH DE number 569384