Global solutions of abstract quasi-linear evolution equations of ``hyperbolic'' type
The author considers the problem \[ u'(t)= A(u(t))u(t)\quad\text{for }t\geq 0,\;u(0)= u_0,\tag{\(*\)} \] where \(\{A(w):w\in Y\}\) is a family of closed linear operators in a real Banach space \(Z\) and \(Y\) is another real Banach space densely and continuously imbedded in \(Z\). Let \(\varphi:Y\to [0,\infty)\) be continuous and bounded on each bounded subset of \(Y\) and such that for each \(\alpha\geq 0\) the set \(Y_\alpha:= \{w\in Y:\varphi(w)\leq\alpha\}\) is bounded in \(Y\). Using the functional \(\varphi\) the author formulates some assumptions in a local sense on the operators \(A(w)\) and proves that for each \(T>0\) the problem \((*)\) has at most one classical solution on \([0,T]\). A continuous function \(g: [0,\infty)\to [0,\infty)\) is called a comparison function if there exists an \(\alpha_0> 0\) such that \(\tau(\alpha_0)= \infty\), where \([0,\tau(\alpha)]\) denotes the interval of existence of the nonextensible maximal solution \(m(t;\alpha)\) to the problem \(r'(t)= g(r(t))\) for \(t\geq 0\), \(r(0)= \alpha\). The main theorem states that if there is a comparison function \(g\) such that \(\lim\inf_{h\downarrow 0}(\varphi((I- hA(w))^{-1}w)- \varphi(w))/h\leq g(\varphi(w))\) for \(w\in Y\), then for each \(u_0\in Y_{\alpha_0}\) there exists the unique global (in time) classical solution \(\widetilde u\) to problem \((*)\) satisfying the growth condition \(\varphi(\widetilde u(t))\leq m(t; \varphi(t_0))\) for \(t\geq 0\). A proof is given by using the following semi-implicit discrete approximations to the problem \((*)\): \((u_i- u_{i- 1})/(t_i- t_{i-1})= A(u_{i- 1})u_i\) for \(i= 1,2,\dots\), with \(0= t_0< t_1< t_2\cdots\). In the last section, the author applies his abstract theory to an abstract version in a real Hilbert space of (1) damped extensible beam equations, (2) of a quasilinear wave equation of Kirchhoff type and to the quasilinear wave system \(\partial_tu= \partial_xv\), \(\partial_tv= \partial_x\delta'(u)- \nu v\), where \(\nu>0\) and \(\delta\in C^5(\mathbb{R})\).
scientific article; zbMATH DE number 1310639