## 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.

### MSC:

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

### Keywords:

nonlinear parabolic systems; nonlinear wave equations