Global existence of classical solutions to the Cauchy problem on a semi-bounded initial axis for a nonhomogeneous quasilinear hyperbolic system

It is proven that if the leftmost eigenvalue is weakly linearly degenerate, then the Cauchy problem for a class of nonhomogeneous quasilinear hyperbolic systems with small and decaying initial data given on a semi-bounded axis admits a unique global C¹ solution on the domain , where x=xn(t) is the fastest forward characteristic emanating from the origin. As an application of our result, we prove the existence of global classical, C¹ solutions of the flow equations of a model class of fluids with viscosity induced by fading memory with small smooth initial data given on a semi-bounded axis.     

Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation

We consider a fourth-order quasilinear nonhomogeneous equation which is equivalent to a nonhomogeneous Hamiltonian system. The purpose of this work is to prove the existence of at least two solutions for such equation when a certain parameter is small enough. Furthermore, under an additional hypothesis on positiveness of the nonhomogeneous part we prove that our solutions are positive.     

Singular nonhomogeneous quasilinear elliptic equations with a convection term

In this work we establish existence results for a class of nonhomogeneous and singular quasilinear elliptic equations involving a convection term. The gradient term makes the problem non variational, and in addition to this difficulty we have to handle the singular term with a sign changing nonlinearity. The proof of the results are made combining the sub‐super solution method, fixed point theorem, Leray–Schauder degree theory and comparison theorems.     

Global classical solutions for a class of quasilinear hyperbolic systems

In this paper we investigate the mechanism of singularity formation to the Cauchy problem of quasilinear hyperbolic system.Moreover, we present some examples to show some significant problems.     

Multiplicity of positive solutions for a class of quasilinear nonhomogeneous Neumann problems

In this paper we study the existence, nonexistence and multiplicity of positive solutions for nonhomogeneous Neumann boundary value problem of the type

where Ω is a bounded domain in with smooth boundary, 1<p<n,Δ_pu=div(|u|^p-2u) is the p-Laplacian operator, , , (x)0 and λ is a real parameter. The proofs of our main results rely on different methods: lower and upper solutions and variational approach.     

Uniqueness of weak solutions to a non-linear hyperbolic system in electrohydrodynamics

In this note we prove the uniqueness of weak solutions to a nonlinear hyperbolic system in electrohydrodynamics without the effects of a dissociation–recombination process. It is still open in the presence of a special dissociation–recombination process, although the existence of at least one weak solution was proved via the method of renormalized solutions by Feireisl [E. Feireisl, Weak solutions to a non-linear hyperbolic system arising in the theory of dielectric liquids, Math. Methods Appl. Sci. 18 (1995) 1041–1052] in 1995.     

Global bounded weak solutions to a degenerate quasilinear chemotaxis system with rotation

This paper deals with the quasilinear Keller–Segel system with rotation where is a bounded domain with smooth boundary, D(u) is supposed to be sufficiently smooth and satisfies D(u)≥D₀u^{m ? 1}(m≥1) and D(u)≤D₁(u + 1)^{K ? m}u^{m ? 1}(K≥1) for all u≥0 with some positive constants D₀ and D₁, and f(u) is assumed to be smooth enough and non‐negative for all u≥0 and f(0) = 0, while S(u,v,x) = (s_ij)_{n × n} is a matrix with and with l≥2, where is nondecreasing on [0,∞). It is proved that when , the system possesses at least one global and bounded weak solution for any sufficiently smooth non‐negative initial data. Copyright © 2015 John Wiley & Sons, Ltd.     

Periodic solutions of second-order quasilinear hyperbolic equations

We study a periodic boundary-value problem for a quasilinear equation with the d'Alembert operator on the left-hand side and a nonlinear operator on the right-hand side and establish conditions under which the solution of the indicated problem is unique.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1370–1375, October, 1995.     

Continuity of solutions of a quasilinear hyperbolic equation with hysteresis

This paper is devoted to the investigation of quasilinear hyperbolic equations of first order with convex and nonconvex hysteresis operator. It is shown that in the nonconvex case the equation, whose nonlinearity is caused by the hysteresis term, has properties analogous to the quasilinear hyperbolic equation of first order. Hysteresis is represented by a functional describing adsorption and desorption on the particles of the substance. An existence result is achieved by using an approximation of implicit time discretization scheme, a priori estimates and passage to the limit; in the convex case it implies the existence of a continuous solution.     

Global and blow-up solutions for a quasilinear hyperbolic equation with strong damping

In this paper, we study a quasilinear hyperbolic equation with strong damping. Firstly, by use of the successive approximation method and a series of classical estimates, we prove the local existence and uniqueness of a weak solution. Secondly, via some inequalities, the potential method and the concave method, we derive the asymptotic and blow-up behavior of the weak solution with different conditions.     

Global weak solution for a symmetrically hyperbolic system

In this work, the existence of bounded weak solutions is obtained for the Cauchy problem of a symmetrically hyperbolic system, arising in such areas as elasticity theory, magnetohydrodynamics, and enhanced oil recovery.     

BREAKDOWN OF CLASSICAL SOLUTIONS TO QUASILINEAR HYPERBOLIC SYSTEMS

This paper deals with the asymptotic behavior of the life-span of classical solutions to Cauchy problem for general first order quasilinear strictly hyperbolic systems in two independent variables with weaker decaying initial data, and obtains a blow-up result for C1 solution to Cauchy problem.