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Abstract: This paper concerns the Cauchy problem for homogeneous weakly hyperbolic equations with time depending analytic coefficients. We give a sufficient condition for the -well-posedness which is also necessary if the space dimension is equal to one. The main point of the paper consists in expressing our condition only in terms of the coefficients of the operator, without needing to know the behavior of the characteristic roots. This is made possible by using the so-called standard symmetrizer of a companion hyperbolic matrix. [Copyright &y& Elsevier]

We consider the Cauchy problem for a multidimensional scalar conservation law and construct an outer estimate for the domain of dependence of its Kružkov solution. The estimate can be represented as the controllability set of a specific differential inclusion. In addition, reachable sets of this inclusion provide outer estimates for the support of the wave profiles. Both results follow from a modified version of the classical Kružkov uniqueness theorem, which we also present in the paper. Finally, the results are applied to a control problem consisting in steering a distributed quantity to a given set. [ABSTRACT FROM AUTHOR]

In this paper, we find that the unbounded spaces Y σ ( R N ) ( 0 < σ < 2 m − 1 ) can provide the work spaces where complicated asymptotic behavior appears in the solutions of the Cauchy problem of the porous medium equation. To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions, we establish the propagation estimates, the growth estimates and the weighted L 1 – L ∞ estimates for the solutions. [ABSTRACT FROM AUTHOR]

In this paper we consider an energy critical wave equation ( 3 ≤ d ≤ 5 , ζ = ± 1 ) ∂ t 2 u − Δ u = ζ ϕ ( x ) | u | 4 / ( d − 2 ) u , ( x , t ) ∈ R d × R with initial data ( u , ∂ t u ) | t = 0 = ( u 0 , u 1 ) ∈ H ˙ 1 × L 2 ( R d ) . Here ϕ ∈ C ( R d ; ( 0 , 1 ] ) converges as | x | → ∞ and satisfies certain technical conditions. We generalize Kenig and Merle's results on the Cauchy problem of the equation ∂ t 2 u − Δ u = | u | 4 / ( d − 2 ) u . Following a similar compactness-rigidity argument we prove that any solution with a finite energy must scatter in the defocusing case ζ = − 1 . While in the focusing case ζ = 1 we give a criterion for global behaviour of the solutions, either scattering or finite-time blow-up when the energy is smaller than a certain threshold. As an application we give a similar criterion on the global behaviour of radial solutions to the focusing, energy critical shifted wave equation ∂ t 2 v − ( Δ H 3 + 1 ) v = | v | 4 v on the hyperbolic space H 3 . [ABSTRACT FROM AUTHOR]

In this paper, we prove the solution map of the Cauchy problem of Camassa–Holm type equations depends continuously on the initial data in nonhomogeneous Besov spaces in the sense of Hadamard by using the Littlewood–Paley theory and the method introduced by Kato [37] and Danchin [21] . [ABSTRACT FROM AUTHOR]

This paper is devoted to an integrable two-component Camassa–Holm system with cubic nonlinearity, which includes the cubic Camassa–Holm equation (also called the Fokas–Olver–Rosenau–Qiao equation) as a special case. The one peaked solitons (peakons) and two peakon solutions are described in an explicit formula. Then, the local well-posedness for the Cauchy problem of the system is studied. Moreover, we target at the precise blow-up scenario for strong solutions to the system, and establish a new blow-up result with respect to the initial data. [ABSTRACT FROM AUTHOR]

Abstract: This paper is concerned with the Cauchy problem of a model equation for shallow water waves of moderate amplitude, which was proposed by A. Constantin and D. Lannes [The hydrodynamical relevance of the Camassa–Holmand Degasperis–Procesi equations, Arch. Ration. Mech. Anal. 192 (2009) 165–186]. First, the local well-posedness of the model equation is obtained in Besov spaces , , (which generalize the Sobolev spaces ) by using Littlewood–Paley decomposition and transport equation theory. Second, the local well-posedness in critical case (with , , ) is considered. Moreover, with analytic initial data, we show that its solutions are analytic in both variables, globally in space and locally in time. Finally, persistence properties on strong solutions are also investigated. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the Cauchy problem for a two-component Degasperis–Procesi system. Firstly, the local well-posedness for this system in the nonhomogeneous Besov spaces is established. Then the precise blow-up scenario for strong solutions to the system is derived. Finally, two new blow-up criterions and the exact blow-up rate of strong solutions to the system are presented. [Copyright &y& Elsevier]

Abstract: This paper is concerned with the large time behavior of solutions to a radiating gas model, which is represented mathematically as a Cauchy problem for a one-dimensional hyperbolic–elliptic coupled system, with suitably given far field states. Suppose the corresponding Riemann problem for the Euler system admits a contact discontinuity wave, then we can construct a “viscous contact wave” for such a hyperbolic–elliptic system. Based on the energy methods and the ellipticity of the radiation flux equation, we prove that the “viscous contact wave” is asymptotically stable provided that the strength of contact discontinuity and the perturbation of the initial data are suitably small. [Copyright &y& Elsevier]

Abstract: In this paper we consider nonlinear boundary value problems whose simplest model is the following: where is a summable function in Ω (bounded open set in , ), , and . [Copyright &y& Elsevier]