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Displaying similar documents to “The initial value problem for the quadratic nonlinear Klein-Gordon equation.”

On the hierarchies of higher order mKdV and KdV equations

Axel Grünrock (2010)

Open Mathematics

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The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces H ^ s r ℝ defined by the norm v 0 H ^ s r ℝ : = ξ s v 0 ^ L ξ r ' , ξ = 1 + ξ 2 1 2 , 1 r + 1 r ' = 1 . Local well-posedness for the jth equation is shown in the parameter range 2 ≥ 1, r > 1, s ≥ 2 j - 1 2 r ' . The proof uses an appropriate variant of the Fourier restriction norm method. A counterexample is discussed to show that the Cauchy problem for equations of this type is in general ill-posed in the C 0-uniform sense, if s < 2 j - 1 2 r ' . The results for r =...

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

Albert J. Milani, Hans Volkmer (2011)

Applications of Mathematics

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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.