## On potential wells and applications to semilinear hyperbolic equations and parabolic equations.(English)Zbl 1096.35089

The authors consider both parabolic and hyperbolic partial differential equations. Given the initial boundary value problem for semilinear hyperbolic equations of the form
\begin{aligned} u_{tt}-\triangle u=f(u), \quad & x\in \Omega , \;t>0, \\ u(x,0)=u_0(x), \;u_t(x,0)=u_1(x), \quad & x\in \Omega,\\ u(x,t)=0, \quad &x\in \partial\Omega , \;\;t\geq 0 \end{aligned}
and semilinear hyperbolic equations of the form,
\begin{aligned} u_{t}-\triangle u=f(u), \quad & x\in \Omega , \;t>0, \\ u(x,0)=u_0(x), \quad & x\in \Omega ,\\ u(x,t)=0, \quad & x\in \partial\Omega , \;t\geq 0. \end{aligned}
Here $$\Omega$$ is a bounded domain in $$\text{ I\hskip-2pt R}^n$$. The family of potential wells $$W$$ is generalized by $W=\{u\in H_0^1(\Omega )| \mid (u)>0, J(u)<d\}\cup \{0\},$ where $$J(u)=(1/2)\| \nabla u\| ^2-\int_{\Omega}F(u)\,dx$$, $$F(u)=\int_{0}^{u}f(s)\,ds$$, $$I(u)=\| \nabla u\| ^2-\int_{\Omega}uf(u)\,dx$$, $$d=\inf{J(u)}$$, $$u\in H_0^1(\Omega )$$, $$\| \nabla u\| \neq 0$$ and $$I(u)=0$$. They use and generalize some results of L. E. Payne and D. H. Sattinger [Isr. J. Math. 22 (1975), 273–303 (1976; Zbl 0317.35059)]. A threshold result of global existence and nonexistence of solutions is obtained. The vacuum isolating of the solutions is discussed as well. The global existence of the solutions is proved for the critical initial conditions $$I(u_0)\geq 0$$, $$E(0)=d$$ (the energy in the initial time) or $$I(u_0)\geq 0$$, $$J(u_0)= d$$.

### MSC:

 35L70 Second-order nonlinear hyperbolic equations 35K55 Nonlinear parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations

Zbl 0317.35059
Full Text:

### References:

 [1] Ball, J. M., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math., 28, 473-486 (1977) · Zbl 0377.35037 [2] Cavalcanti, M. M.; Domingos Cavalcanti, V. N.; Martinez, P., Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203, 119-158 (2004) · Zbl 1049.35047 [3] Cavalcanti, M. M.; Domingos Cavalcanti, V. N., Existence and asymptotic stability for evolution problems on manifolds with damping and source terms, J. Math. Anal. Appl., 291, 109-127 (2004) · Zbl 1073.35168 [4] Esquivel-Avila, J. A., A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations, Nonlinear Anal., 52, 1111-1127 (2003) · Zbl 1023.35076 [5] Esquivel-Avila, J. A., The dynamics of a nonlinear wave equation, J. Math. Anal. Appl., 279, 135-150 (2003) · Zbl 1015.35072 [6] Esquivel-Avila, J. A., Qualitative analysis of a nonlinear wave equation, Discrete Continuous Dynam. Syst., 10, 3, 787-804 (2004) · Zbl 1047.35103 [7] J.A. Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal., in press.; J.A. Esquivel-Avila, Dynamics around the ground state of a nonlinear evolution equation, Nonlinear Anal., in press. · Zbl 1159.35390 [8] Gan, Z.; Zhang, J., Instability of standing waves for Klein-Gordon-Zakharov equations with different propagation speeds in three space dimensions, J. Math. Anal. Appl., 307, 219-231 (2005) · Zbl 1068.35066 [9] F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. I. H. Poincar $$e$$; F. Gazzola, M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. I. H. Poincar $$e$$ · Zbl 1094.35082 [10] Ikehata, R., Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27, 1165-1175 (1996) · Zbl 0866.35071 [11] Lions, J. L., Quelques methods de resolution des problem aux limits nonlinears (1969), Dunod: Dunod Paris · Zbl 0189.40603 [12] L. Liu, M. Wang, Global solutions and blow-up of solutions for some hyperbolic systems with damping and source terms, Nonlinear Anal., in press.; L. Liu, M. Wang, Global solutions and blow-up of solutions for some hyperbolic systems with damping and source terms, Nonlinear Anal., in press. · Zbl 1082.35100 [13] Nakao, M.; Ono, K., Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations, Math. Z., 214, 2, 325-342 (1993) · Zbl 0790.35072 [14] Ono, K., On global existence, asymptotic stability and blowing up of solutions for some degenerate non-linear wave equations of Kirchhoff type with a strong dissipation, Math. Methods Appl. Sci., 20, 2, 151-177 (1997) · Zbl 0878.35081 [15] Payne, L. E.; Sattinger, D. H., Sadle points and instability of nonlinear hyperbolic equations, Israel. J. Math., 22, 273-303 (1975) · Zbl 0317.35059 [16] Sattinger, D. H., On global solution of nonlinear hyperbolic equations, Arch. Rat. Mech. Anal., 30, 148-172 (1968) · Zbl 0159.39102 [17] Tsutsumi, M., On solutions of semilinear differential equations in a Hilbert space, Math. Japan, 17, 173-193 (1972) · Zbl 0273.34044 [18] Tsutsumi, M., Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. RTMS, 8, 211-229 (1972/73) · Zbl 0248.35074 [19] Vitillaro, E., A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasgow Math. J., 44, 3, 375-395 (2002) · Zbl 1016.35048 [20] Yacheng, L., On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192, 155-169 (2003) · Zbl 1024.35078 [21] Yacheng, L.; Junsheng, Z., Multidimensional viscoelasticity equations with nonlinear damping and source terms, Nonlinear Anal., 56, 851-865 (2004) · Zbl 1057.74007 [22] Yacheng, L.; Junsheng, Z., Nonlinear parabolic equations with critical initial conditions $$J(u_0) = d$$ or $$I(u_0) = 0$$, Nonlinear Anal., 58, 873-883 (2004) · Zbl 1059.35064 [23] Zhang, J., On the standing wave in coupled non-linear Klein-Gordon equations, Math. Methods Appl. Sci., 26, 1, 11-25 (2003) · Zbl 1034.35080
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.