zbMATH — the first resource for mathematics

Weak solutions for a system of nonlinear Klein-Gordon equations. (English) Zbl 0633.35053
Let \(n\geq 3\) be the dimension of \(R^ n\). Let us consider a real number \(\rho\) satisfying the following condition: \[ (1)\quad -1<\rho <4/(n-2). \] Let \(\theta\) and \(\gamma\) be the following real numbers: \[ (2)\quad \theta =2n(\rho +2)/((n-2)(\rho +2)+2n(\rho +1)),\quad \gamma =2n(\rho +2)/((n+2)(\rho +2)-2n(\rho +1)). \] Clearly, \(1/\theta +1/\gamma =1\) and \[ (3)\quad 1<\theta <(\rho +2)/(\rho +1),\quad \gamma >1. \] Then the authors prove:
Theorem 1. Let \(\Omega\) be a regular bounded domain of \(R^ n\) and \(\rho\) a real number satisfying the condition (1) if \(n\geq 3\) or \(\rho >- 1\) if \(n=1,2\). Let \[ (4)\quad f_ 1,f_ 2\in L^ 2(0,T;L^ 2(\Omega)), \] \[ (5)\quad u_ 0,v_ 0\in H^ 1_ 0(\Omega)\cap L^ p(\Omega), \] \[ (6)\quad u_ 1,v_ 1\in L^ 2(0,T;L^ 2(\Omega)), \] where \(p=\rho +2\). Then there exists functions u,v: ]0,T[\(\to L^ 2(\Omega)\) such that: \[ (7)\quad u,v\in L^{\infty}(0,T;H^ 1_ 0(\Omega)), \] \[ (8)\quad u',v'\in L^{\infty}(0,T;L^ 2(\Omega))\quad (u'=du/dt), \] \[ (9)\quad uv\in L^{\infty}(0,t;L^{\rho +2}(\Omega)), \] satisfying the nonlinear systems: \[ (10)\quad u''-\Delta u+| v|^{\rho +2} | u|^{\rho} u=f_ 1\quad in\quad L^ 2(0,T;H^{-1}(\Omega)+L^{\theta}(\Omega)), \] \[ (11)\quad v''-\Delta v+| u|^{\rho +2} | v|^{\rho} v=f_ 2\quad in\quad L^ 2(0,T;H^{-1}(\Omega)+L^{\theta}(\Omega)); \] and the initial conditions: \[ (12)\quad u(0)=u_ 0,\quad v(0)=v_ 0;\quad (13)\quad u'(0)=u_ 1,\quad v'(0)=v_ 1. \] Theorem 2. Let u,v: ]0,T[\(\to L^ 2(\Omega)\) be functions in the classes (7), (8) and (9) satisfying from (10) to (13). Then, \(u=v\) provided that \(\rho\geq 0\) in case \(n=1\) or 2; \(u=v\) if \(\rho =0\) in case \(n=3\).
Reviewer: Y.Ebihara

35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
Full Text: DOI
[1] J.Ferreira - G.Perla Menzala,Decay of solutions of a system of nonlinear Klein-Gordon equations (to appear). · Zbl 0617.35073
[2] K.Jörgens,Nonlinear wave equations, University of Colorado, Department of Mathematics, 1970.
[3] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Paris: Dunod, Paris · Zbl 0189.40603
[4] Lions, J. L.; Magenes, E., Problèmes aux limites non homogènes et applications, Vol. 1 (1968), Paris: Dunod, Paris · Zbl 0165.10801
[5] Lions, J. L.; Strauss, W. A., Some non linear evolutions equations, Bull. Soc. Math. de France, 95, 43-96 (1965) · Zbl 0132.10501
[6] Makhankov, V. G., Dynamics of classical solutions in integrable systems, Physics Reports (Section C of Physics Letters), 35, 1, 1-128 (1978)
[7] L. A.Medeiros - G.Perla Menzala,On a mixed problem for a class of nonlinear Klein-Gordon equations (to appear). · Zbl 0682.35072
[8] Segal, I., Nonlinear partial differential equations in Quantum Field Theory, Proc. Symp. Appl. Math. A.M.S., 17, 210-226 (1965)
[9] M. I.Visik - O. A.Ladyzhenskata,On boundary value problems for partial differential equations and certain class of operator equations, A.M.S. Translations Series 2, vol. 10, 1958.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.