Remarks on blowup of solutions for nonlinear evolution equations of second order. (English) Zbl 0920.35025

Let \(H\) be real Hilbert space with the norm \(| \cdot| \) and the inner product \((\cdot,\cdot)\) respectively, and \(V\) is the real Banach space with the continuous injection \(V\to H\). The author studies the problem of non-existence of global solutions for the following nonlinear evolution equation \[ u''(t)+Bu'(t)+J'(u(t))=0,\qquad t\geq 0 \tag{1} \]
\[ u(0)=u_0, u'(0)=u_1, \tag{2} \] where \(J'\) denotes the derivative of \(J\), and \(B\) is a non-negative selfadjoint operator in \(H\). For \(u\in V\) and \(v\in H (J(0)=0)\) set \(K(u)=\langle u,J'(u)\rangle\), \(E(u,v)=\frac{1}{2} | v| ^2_H+J(u)\) where \(\langle\cdot, \cdot\rangle\) denotes the duality between \(V\) and the dual space \(V'\). Denote the potential depth \(d\) and the unstable set \({\mathcal V}\) as follows, \(d=\inf\{J(u)\); \(u\in V\), \(K(u)=0\), \(u\neq 0\}\), \({\mathcal V}=\{(u,v) \in V\times H\); \(E(u,v)<d\), \(K(u)<0\}\). The function \(u(t)\) is called a weak solution of the problem (1),(2) in \([0,T]\) if \(u\in C([0,T]\); \(V\cap D(B^{\frac{1}{2}}))\), \(u'(t)\in C([0,T];H)\cap L^2([0,T];D(B^{\frac{1}{2}}))\) and \(u\) satisfies \((2)\) and \[ \frac{d}{dt}(u(t),u'(t))_H +(B^{\frac{1}{2}}u(t),B^{\frac{1}{2}}u'(t))_H\geq | u'(t)| ^2_H- K(u(t)), \]
\[ E(u(t),u'(t))+\int _0^t | B^{\frac{1}{2}}u'(s)| ^2_H ds\leq E(u_0,u_1), \] for almost all \(t\in [0,T]\).
Theorem. Assume that there exists a constant \(q>2\) such that \(qJ(u)-K(u)\geq 0\) for all \(u\in V\) and that \(d=\inf\{J(u)-1/qK(u)\); \(u\in V\), \(K(u)\leq 0, u\neq 0\}\). Then any weak solution \(u(t)\) of the problem (1),(2) satisfying \((u_0,u_1) \in {\mathcal V}\) and \(u_0\in D(B^{\frac{1}{2}})\) does not exist globally in time.
This theorem has applications to the non-existens problem of global solutions to the problem \[ u_{tt}-\Delta u_t -(\alpha+\beta\| \nabla u\| ^2_2)\Delta u - | u| ^{q-2} u=0, \quad x \in \Omega, \]
\[ u(t,x)=0, \quad x\in \partial \Omega, \] and also to the problem \[ u_t=d_1\Delta u+| u-v| ^{q-2}(u-v)-u,\;v_t=d_2\Delta v+| u-v| ^{q-2}(u-v)-v,\quad x\in\Omega, \]
\[ (\partial u/\partial\nu)(t,x)=(\partial v/\partial\nu)(t,x)=0,\quad x\in \partial \Omega, \]
\[ u(0,x)=u_0(x),\quad v(0,x)=v_0(x),\qquad x\in \Omega, \] where \(d_1,d_2\geq 0, q>2\), and \(\Omega\subset \mathbb{R}^n\) is bounded with smooth boundary.


35B40 Asymptotic behavior of solutions to PDEs
34G20 Nonlinear differential equations in abstract spaces
47H20 Semigroups of nonlinear operators