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Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

Albert J. Milani;  Hans Volkmer — 2011

Applications of Mathematics

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation u t t + 2 u t - a i j ( u t , ∇ u ) ∂ i ∂ j u = f corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation - a i j ( 0 , ∇ v ) ∂ i ∂ j v = h . We then give conditions for the convergence, as t → ∞ , of the solution of the evolution equation to its stationary state.

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