Blow-up of solutions for some non-linear wave equations of Kirchhoff type with some dissipation

The initial boundary value problem for non-linear wave equations of Kirchhoff type with dissipation in a bounded domain is considered. We prove the blow-up of solutions for the strong dissipative term -Δ_ut$-\mathrm{\Delta }{u}_t$ and the linear dissipative term _ut$u_t$ by the energy method and give some estimates for the life span of solutions. We also show the nonexistence of global solutions with positive initial energy for non-linear dissipative term by Vitillaro's argument.     

Hyperbolic-parabolic singular perturbation for quasilinear equations of Kirchhoff type

We consider a hyperbolic-parabolic singular perturbation problem for a quasilinear equation of Kirchhoff type, and obtain parameter-dependent time decay estimates of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation. For this purpose we show time decay estimates for hyperbolic-parabolic singular perturbation problem for linear equations with a time-dependent coefficient.     

Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions

We study the existence of (generalized) bounded solutions existing for all times for nonlinear parabolic equations with nonlinear boundary conditions on a domain that is bounded in space and unbounded in time (the entire real line). We give a counterexample which shows that a (weak) maximum principle does not hold in general for linear problems defined on the entire real line in time. We consider a boundedness condition at minus infinity to establish (one-sided) L^∞-a priori estimates for solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. We then take up the case of nonlinear problems with (possibly) nonlinear boundary conditions. By using comparison techniques, some (delicate) a priori estimates obtained herein, and nonlinear approximation methods, we prove the existence and, in some instances, positivity and uniqueness of strong full bounded solutions existing for all times.     

On sharp decay estimates of solutions for mildly degenerate dissipative wave equations of Kirchhoff type

We consider the initial data boundary value problem for the degenerate dissipative wave equations of Kirchhoff type ρu′′ + ∥A^1/2u∥^2γAu+ u′ = 0. When either the coefficient ρ or the initial data are appropriately small at least, we show the global existence theorem by using suitable identities together with the energy. Moreover, under the same assumption for ρ and the initial data, we derive the sharp decay estimates of the solutions and their second derivatives. Copyright © 2011 John Wiley & Sons, Ltd.     

The mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant

In this paper, we investigate the mixed initial-boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half-unbounded domain . Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C¹ solutions with the bounded L¹∩L^∞ norm of the initial data as well as their derivatives and appropriate boundary condition. Based on the existence results of global classical solutions, we also prove that when t tends to infinity, the solutions approach a combination of C¹ travelling wave solutions. Under the appropriate assumptions of initial and boundary data, the results can be applied to the equation of time-like extremal surface in Minkowski space R^1+(1+n).     

Higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance (UTSD) equations

We establish C^2,α$C^{2,\alpha }$-estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems. Using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance (UTSD) equation. We also present Lipschitz estimates near the degenerate Dirichlet boundary (the sonic boundary) for the UTSD equation.     

Global classical large solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum

In this paper, we investigate an initial boundary value problem for 1D compressible isentropic Navier-Stokes equations with large initial data, density-dependent viscosity, external force, and vacuum. Making full use of the local estimates of the solutions in Cho and Kim (2006) [3] and the one-dimensional properties of the equations and the Sobolev inequalities, we get a unique global classical solution (ρ,u) where ρ∈C¹([0,T];H¹([0,1])) and u∈H¹([0,T];H²([0,1])) for any T>0. As it is pointed out in Xin (1998) [31] that the smooth solution (ρ,u)∈C¹([0,T];H³(R¹)) (T is large enough) of the Cauchy problem must blow up in finite time when the initial density is of nontrivial compact support. It seems that the regularities of the solutions we obtained can be improved, which motivates us to obtain some new estimates with the help of a new test function ρ²utt, such as Lemmas 3.2-3.6. This leads to further regularities of (ρ,u) where ρ∈C¹([0,T];H³([0,1])), u∈H¹([0,T];H³([0,1])). It is still open whether the regularity of u could be improved to C¹([0,T];H³([0,1])) with the appearance of vacuum, since it is not obvious that the solutions in C¹([0,T];H³([0,1])) to the initial boundary value problem must blow up in finite time.     

Second-order type-changing evolution equations with first-order intermediate equations

This paper presents a partial classification for C^∞ type-changing symplectic Monge-Ampère partial differential equations (PDEs) that possess an infinite set of first-order intermediate PDEs. The normal forms will be quasi-linear evolution equations whose types change from hyperbolic to either parabolic or to zero. The zero points can be viewed as analogous to singular points in ordinary differential equations. In some cases, intermediate PDEs can be used to establish existence of solutions for ill-posed initial value problems.     

Local solutions for a coupled system of Kirchhoff type

We investigate the existence of local solutions of the following coupled system of Kirchhoff equations subject to nonlinear dissipation on the boundary: (∗) Here {Γ₀,Γ₁} is an appropriate partition of the boundary Γ of Ω and ν(x), the outer unit normal vector at x∈Γ₁.By applying the Galerkin method with a special basis for the space where lie the approximations of the initial data, we obtain local solutions of the initial-boundary value problem for (∗).     

Characteristic boundary layers for parabolic perturbations of quasilinear hyperbolic problems

In this paper, we study the existence and nonlinear stability of the totally characteristic boundary layer for the quasilinear equations with positive definite viscosity matrix under the assumption that the boundary matrix vanishes identically on the boundary x=0. We carry out a series of weighted estimates to the boundary layer equations—Prandtl type equations to get the regularity and the far field behavior of the solutions. This allows us to perform a weighted energy estimate for the error equation to prove the stability of the boundary layers. The stability result finally implies the asymptotic limit of the viscous solutions.     

Large time behavior and L−L estimate of solutions of 2-dimensional nonlinear damped wave equations

We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the L^p−L^q type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the L^p−L^q estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity |u|^αu. Our result covers the whole super critical case α>1, where the α=1 is well known as the Fujita exponent when n=2.     

Nonlinear analysis on the vibration of elastic plates

We consider the vibration of elastic thin plates under certain reasonable assumptions. We derive the nonlinear equations for this model by the Hamilton Principle. Under the conditions on the hyperbolicity for the initial data, we establish the local time well-posedness for the initial and boundary value problem by Picard iteration scheme, and obtain the estimates for the solutions.     

Global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields

In this paper, we study the global existence and the asymptotic behavior of classical solution of the Cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields. We prove that the global C¹ solution exists uniquely if the BV norm of the initial data is sufficiently small. Based on the existence result on the global classical solution, we show that, when the time t tends to the infinity, the solution approaches a combination of C¹ traveling wave solutions. Finally, we give an application to the equation for time-like extremal surfaces in the Minkowski space-time R1+n.     

Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems

In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant . Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence and uniqueness of C¹ solutions are obtained for small initial and boundary data. We also present two applications for physical models.     

Global solutions and self-similar solutions of semilinear parabolic equations with nonlinear gradient terms

In this paper we consider the Cauchy problem of semilinear parabolic equations with nonlinear gradient terms a(x)|u|^q−1u|∇u|^p. We prove the existence of global solutions and self-similar solutions for small initial data. Moreover, for a class of initial data we show that the global solutions behave asymptotically like self-similar solutions as t→∞.     

Initial-boundary problem for the 1-D Euler-Boltzmann equations in radiation hydrodynamics

We study the initial-boundary value problem for the one dimensional EulerBoltzmann equation with reflection boundary condition. For initial data with small total variation, we use a modified Glimm scheme to construct the global approximate solutions(U_(△t,d), I_(△t,d)) and prove that there is a subsequence of the approximate solutions which is convergent to the global solution.     

Global existence for Maxwell-Bloch systems

Maxwell-Bloch equations describe the propagation of an electromagnetic wave through a quantum medium. For any number of quantum levels, in space dimension 3, we show the global existence of weak (L²) solutions to the initial-value problem. In the case of smoother electromagnetic fields (with curl in L²), the solution is unique. For smooth data (H^s, s?2), the solutions remain smooth for all times.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

Scalar multidimensional conservation laws IBVP in noncylindrical Lipschitz domains

We study the initial-boundary-value problems for multidimensional scalar conservation laws in noncylindrical domains with Lipschitz boundary. We show the existence-uniqueness of this problem for initial-boundary data in L^∞ and the flux-function in the class C¹. In fact, first considering smooth boundary, we obtain the L¹-contraction property, discuss the existence problem and prove it by the Young measures theory. In the end we show how to pass the existence-uniqueness results on to some domains with Lipschitz boundary.     

ALMOST OPTIMAL LOCAL WELL-POSEDNESS OF THE MAXWELL-KLEIN-GORDON EQUATIONS IN 1 + 4 DIMENSIONS

Abstract

We study strong solutions of the Navier–Stokes equations for nonhomogeneous incompressible fluids in Ω ? R ³. Deriving higher a priori estimates independent of the lower bounds of the density, we prove the existence and uniqueness of local strong solutions to the initial value problem (for Ω =R ³) or the initial boundary value problem (for Ω ? ? R ³) even though the initial density vanishes in an open subset of Ω, i.e., an initial vacuum exists. As an immediate consequence of the a priori estimates, we obtain a continuation theorem for the local strong solutions.