Parametric resonance and nonexistence of global solution to nonlinear hyperbolic equations (Q2704054)

The author's starting point is the nonlinear Cauchy problem NEWLINE\[NEWLINEu_{tt}-\Delta u= f(u_t, \nabla_xu, \nabla_xu_t, \nabla^2_x u),\quad u(0,x)= u_0(x),\quad u_t(0, x)= u_1(x).NEWLINE\]NEWLINE He recalls some conditions which guarantee the global existence (in time) of small data solutions as e.g. relations between the growth behaviour of \(f\) in a neighbourhood of \(0\) and the spatial dimension \(n\). The author is interested in the influence of time-dependent ``coefficients'' on global existence of small data solutions. For this reason he studies the model problem NEWLINE\[NEWLINEu_{tt}- \lambda(t)^2 b(t)^2\Delta u= f(u_t, \lambda(t) b(t)\nabla_x u),\quad u(1,x)= u_0(x),\quad u_t(1,x)= u_1(x)NEWLINE\]NEWLINE with \(\lambda(t):= \exp(t^\alpha)\), \(\alpha\in \mathbb{R}\); \(b= b(t)\) is a periodic, non-constant, smooth and positive function and \(f(s_1, s_2)=|s_2|^2- s^2_1\).NEWLINENEWLINENEWLINEHe obtains an interesting result, that for \(\alpha\in (-\infty,-1)\) (coefficient stabilizes to a periodic one for \(t\to\infty\) in a suitable way) and for \(\alpha= 0\) (pure periodic coefficient) one cannot expect a global existence result of small data solutions. Main tools to prove this result are Nirenberg's transformation \(v= \exp u\) which reduces the nonlinear model problem to a linear one and Floquet theory for corresponding ordinary differential equations.NEWLINENEWLINENEWLINESome open questions motivate the interesting reader to further studies.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00030].