On the exponential growth of solutions to nonlinear hyperbolic equations (Q1826032)

The authors consider the equation \(u_{xt}(x,t)=F(x,t,u(x,t),u_ x(x,t))\) with the initial and boundary conditions \(u(0,t)=u_ 0(t)\), \(0\leq t\leq T\), \(u(x,0)=\phi (x)\), \(0\leq x\leq \ell\), where \(u_ 0(t)\) and \(\phi\) (x) are given functions on [0,\(\ell]\times [0,T]\). The first result is an existence and uniqueness theorem for continuous solutions under the assumption that F satisfies a certain Lipschitz condition. For the equation \(u_{xt}(x,t)+a(x,t)u_ x(x,t)=f(x,t,u(x,t))\) the next two theorems indicate certain conditions which ensure that the solution cannot grow to infinity faster than exponentially in the case of an unbounded region \((T=\infty)\).