## Some remarks on the wave equations with nonlinear damping and source terms.(English)Zbl 0866.35071

The author studies the following mixed problem $u_{tt}- \Delta u+ \delta|u_t |^{m-1} u_t= |u|^{p-1}u, \quad x\in\Omega,\;t\geq 0, \tag{1}$
$u(0,x)= u_0(x),\;u_t(0,x) =u_1(x),\;x\in\Omega, \quad u(t,x)= 0,\;x\in\partial \Omega,\;t\geq 0.$ Here $$\Omega \subset\mathbb{R}^N$$ is a bounded domain with smooth boundary $$\partial\Omega$$, $$p>1$$, $$m\geq 1$$, $$\delta>0$$ and $$\Delta$$ is the Laplacian in $$\mathbb{R}^N$$. The maximal time interval of existence of the solution is such that (1) admits a solution $$u(t,\cdot) \in C([0,T_m); H^1_0 (\Omega)) \cap C^1([0,T_m); L^2(\Omega))$$ with $$u_t\in L^\infty ((0,T)\times \Omega)$$ for any $$0<T< T_m$$ and if $$T_m<\infty$$, then $\lim_{t\nearrow T_m} \bigl|\nabla u(t,\circ) \bigr|_2+ \bigl|u_t(t,\circ) \bigr|_2= \infty.$ To study the existence of a stable set for this problem, the author introduces the notion of potential depth as follows: $J(u)= {1\over 2} |\nabla u |^2_2- {1 \over p+1} |u |^{p+1}_{p+1},$ $$E(u,v)= {1\over 2} |v|^2_2 +J(u)$$, $$I(u)= |\nabla u |^2_2-|u |^{p+1}_{p+1}$$, $$d\equiv \inf\{\sup_{\lambda \geq 0} J(\lambda u)$$; $$u\in H^1_0 (\Omega),\;u\neq 0\}$$. It is well known that the potential depth $$d$$ is positive. The stable set is then defined by $$W^*= \{u\in H^1_0(\Omega)$$; $$J(u)<d$$, $$I(u)>0\} \cup \{0\}$$. The first result is the following.
Theorem 1. Suppose $$\delta>0$$, $$m\geq 1$$ and either $$p>1$$ $$(N=1,2)$$ or $$1<p \leq N/(N-2)$$. Let $$u(t,x)$$ be a local solution of (1) and $$[0,T_m)$$ the maximal time-interval of existence of the solution. If there exists a real number $$t_0\in [0,T_m)$$ such that $$u(t_0,\circ)\in W^*$$ and $$E(u(t_0,\circ), u_t(t_0,\circ)<d$$, then $$T_m= \infty$$.
Under the additional assumptions that either $$m\geq 1$$ $$(N=1,2)$$ or $$1\leq m\leq {N+2 \over N-2}$$ $$(N\geq 3)$$ the author shows that the following two conditions are equivalent:
i) There exists a real number $$t_0\in[0,T_m)$$ such that $$u(t_0,\circ)\in W^*$$ and $$E(u(t_0,\circ)$$, $$u_t [t_0,\circ))<d$$,
ii) $$T_m = \infty$$ and $$\lim_{t\to\infty} |\nabla u(t,\circ)|_2=\lim_{t\to\infty} |u_t(t,\circ) |_2=0$$.
Reviewer: V.Georgiev (Sofia)

### MSC:

 35L70 Second-order nonlinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations
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### References:

 [1] IKEHATA R. & SUZUKI T., Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima math. J. (to appear).; IKEHATA R. & SUZUKI T., Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima math. J. (to appear). · Zbl 0873.35010 [2] Sattinger, D. H., On global solution of nonlinear hyperbolic equations, Archs ration. Mech. Analysis, 30, 148-172 (1968) · Zbl 0159.39102 [3] Payne, L. E.; Sattinger, D. H., Saddle points and unstability of nonlinear hyperbolic equations, Israel J. Math., 22, 273-303 (1975) · Zbl 0317.35059 [4] Georgiev, V.; Todorova, G., Existence of a solution of the wave equation with nonlinear damping and source terms, J. diff. Eqns, 109, 295-308 (1994) · Zbl 0803.35092 [5] Haraux, A., Nonlinear vibrations and the wave equation, (Lecture notes (1986), Université de Paris VI: Université de Paris VI New York) · Zbl 0782.35042 [6] Haraux, A., Semi-linear Hyperbolic Problems in Bounded Domains, (Mathematical Reports, Vol. 3 (1987), Harwood Academic Publishers: Harwood Academic Publishers Switzerland), Part 1 · Zbl 0681.35058 [7] Tsutsumi, M., On solutions of semilinear differential equations in a Hilbert space, Math. Japan., 17, 173-193 (1972) · Zbl 0273.34044 [8] Levine, H. A., Instability and nonexistence of global solutions to nonlinear wave equations of the form $$Pu_{tt} = −Au$$ + ℱ(u), Trans. Am. math. Soc., 192, 1-21 (1974) · Zbl 0288.35003 [9] Lions, J.-L., Quelques méthodes de résolution des problèmes aux limites non lineaires (1969), Dunod: Dunod Switzerland · Zbl 0189.40603 [10] Ôtani, M., Existence and asymptotic stability of strong solutions of nonlinear evolution equations with a difference term of subdifferentials, (Colloq. Math. Soc. Janos Bolyai, Qualitative Theory of Differential Equations, Vol. 30 (1980), North-Holland: North-Holland Paris) · Zbl 0506.35075
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