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We study the wellposedness of Cauchy problem for the fourth order nonlinear Schrödinger equations i∂ t u = −ε∆u + ∆ 2 u + P (∂ α x u) |α|2 , (∂ α x ¯ u) |α|2 , t ∈ R, x ∈ R n , where ε ∈ {−1, 0, 1}, n 2 denotes the spatial dimension and P (·) is a polynomial excluding constant and linear terms.
We study a rate-type viscoelastic system proposed by I. which is a 3_3 hyperbolic system with relaxation. As the relaxation time tends to zero, this system convergences to the well-known p-system formally. In the case where the initial data are the Riemann data such that the corresponding solutions of the p-system are centered rarefaction waves, we show(More)
We study the initial value problem for the system of compressible adiabatic flow through porous media in the one space dimension with fixed boundary condition. Under the restriction on the oscillations in the initial data, we establish the global existence and large time behavior for the classical solutions via the combination of characteristic analysis and(More)