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Weak solutions for a system of nonlinear Klein-Gordon equations. (English) Zbl 0633.35053

Let $n\ge 3$ be the dimension of ${R}^{n}$. Let us consider a real number $\rho$ satisfying the following condition:

$\left(1\right)\phantom{\rule{1.em}{0ex}}-1<\rho <4/\left(n-2\right)·$

Let $\theta$ and $\gamma$ be the following real numbers:

$\left(2\right)\phantom{\rule{1.em}{0ex}}\theta =2n\left(\rho +2\right)/\left(\left(n-2\right)\left(\rho +2\right)+2n\left(\rho +1\right)\right),\phantom{\rule{1.em}{0ex}}\gamma =2n\left(\rho +2\right)/\left(\left(n+2\right)\left(\rho +2\right)-2n\left(\rho +1\right)\right)·$

Clearly, $1/\theta +1/\gamma =1$ and

$\left(3\right)\phantom{\rule{1.em}{0ex}}1<\theta <\left(\rho +2\right)/\left(\rho +1\right),\phantom{\rule{1.em}{0ex}}\gamma >1·$

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 $n\ge 3$ or $\rho >-1$ if $n=1,2$. Let

$\left(4\right)\phantom{\rule{1.em}{0ex}}{f}_{1},{f}_{2}\in {L}^{2}\left(0,T;{L}^{2}\left({\Omega }\right)\right),$
$\left(5\right)\phantom{\rule{1.em}{0ex}}{u}_{0},{v}_{0}\in {H}_{0}^{1}\left({\Omega }\right)\cap {L}^{p}\left({\Omega }\right),$
$\left(6\right)\phantom{\rule{1.em}{0ex}}{u}_{1},{v}_{1}\in {L}^{2}\left(0,T;{L}^{2}\left({\Omega }\right)\right),$

where $p=\rho +2$. Then there exists functions u,v: ]0,T[$\to {L}^{2}\left({\Omega }\right)$ such that:

$\left(7\right)\phantom{\rule{1.em}{0ex}}u,v\in {L}^{\infty }\left(0,T;{H}_{0}^{1}\left({\Omega }\right)\right),$
$\left(8\right)\phantom{\rule{1.em}{0ex}}{u}^{\text{'}},{v}^{\text{'}}\in {L}^{\infty }\left(0,T;{L}^{2}\left({\Omega }\right)\right)\phantom{\rule{1.em}{0ex}}\left({u}^{\text{'}}=du/dt\right),$
$\left(9\right)\phantom{\rule{1.em}{0ex}}uv\in {L}^{\infty }\left(0,t;{L}^{\rho +2}\left({\Omega }\right)\right),$

satisfying the nonlinear systems:

$\left(10\right)\phantom{\rule{1.em}{0ex}}{u}^{\text{'}\text{'}}-{\Delta }u+{|v|}^{\rho +2}{|u|}^{\rho }u={f}_{1}\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}{L}^{2}\left(0,T;{H}^{-1}\left({\Omega }\right)+{L}^{\theta }\left({\Omega }\right)\right),$
$\left(11\right)\phantom{\rule{1.em}{0ex}}{v}^{\text{'}\text{'}}-{\Delta }v+{|u|}^{\rho +2}{|v|}^{\rho }v={f}_{2}\phantom{\rule{1.em}{0ex}}in\phantom{\rule{1.em}{0ex}}{L}^{2}\left(0,T;{H}^{-1}\left({\Omega }\right)+{L}^{\theta }\left({\Omega }\right)\right);$

and the initial conditions:

$\left(12\right)\phantom{\rule{1.em}{0ex}}u\left(0\right)={u}_{0},\phantom{\rule{1.em}{0ex}}v\left(0\right)={v}_{0};\phantom{\rule{1.em}{0ex}}\left(13\right)\phantom{\rule{1.em}{0ex}}{u}^{\text{'}}\left(0\right)={u}_{1},\phantom{\rule{1.em}{0ex}}{v}^{\text{'}}\left(0\right)={v}_{1}·$

Theorem 2. Let u,v: ]0,T[$\to {L}^{2}\left({\Omega }\right)$ be functions in the classes (7), (8) and (9) satisfying from (10) to (13). Then, $u=v$ provided that $\rho \ge 0$ in case $n=1$ or 2; $u=v$ if $\rho =0$ in case $n=3$.

Reviewer: Y.Ebihara

##### MSC:
 35L70 Nonlinear second-order hyperbolic equations 35L20 Second order hyperbolic equations, boundary value problems 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
##### References:
 [1] J.Ferreira - G.Perla Menzala,Decay of solutions of a system of nonlinear Klein-Gordon equations (to appear). [2] K.Jörgens,Nonlinear wave equations, University of Colorado, Department of Mathematics, 1970. [3] J. L. Lions,Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969. [4] J. L. Lions -E. Magenes,Problèmes aux limites non homogènes et applications, Vol. 1, Dunod, Paris, 1968. [5] J. L. Lions -W. A. Strauss,Some non linear evolutions equations, Bull. Soc. Math. de France,95 (1965), pp. 43–96. [6] V. G. Makhankov,Dynamics of classical solutions in integrable systems, Physics Reports (Section C of Physics Letters),35 (1) (1978), pp. 1–128. · doi:10.1016/0370-1573(78)90074-1 [7] L. A.Medeiros - G.Perla Menzala,On a mixed problem for a class of nonlinear Klein-Gordon equations (to appear). [8] I. Segal,Nonlinear partial differential equations in Quantum Field Theory, Proc. Symp. Appl. Math. A.M.S.,17 (1965), pp. 210–226. [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.