A uniqueness result for a nonlinear hyperbolic equation

In this paper we consider the problem of identifying a coefficient function in the principal part of a nonlinear hyperbolic PDE from overdetermined boundary data of the PDE solution. Assuming that the coefficient function can be written as a linear combination of finitely many ansatz functions, we derive a stability and uniqueness result that is based on an identifiability condition in terms of the initial data as well as on the smallness of the time interval. In doing so, we distinguish between the two- and higher dimensional case where functionals on the boundary can be measured and the one-dimensional situation with only point measurements at one boundary point. Our results also give perspectives to the case of an infinite dimensional coefficient function.     

A blow-up result for a higher-order nonlinear Kirchhoff-type hyperbolic equation

In this work we consider a multi-dimensional higher-order Kirchhoff-type wave equation, with Dirichlet boundary conditions. We establish a blow-up result for certain solutions with positive initial energy.     

A solvability result for a nonlinear weakly hyperbolic equation of second order

In this article we study the well-posedness of the initial value problem for quasi-linear weakly hyperbolic equations of second order. We obtain a sufficient condition for the Cauchy problem to be locally solvable in the class of smooth function.     

A global existence and uniqueness result for a class of hyperbolic operators

In this paper a global existence and uniqueness theorem for the Cauchy problem related to the class of hyperbolic operators with double characteristics \(P=D^2_{t} - D^2_{x_1} - (t+ \lambda - \alpha (x_1))^2 D^2_{x_2},\) depending on the parameter \(\lambda \) in the half-space \(\Omega = \mathbb {R}^2 \times ]0, + \infty [\) is proved. In the first part of the paper, a priori estimates in Sobolev spaces \(W^{r,2}\), with \(r\) negative, have been established. Then, the regularity is studied by using these spaces and by constructing a Green’s function in the half-plane for the operator of the waves.     

A nondegeneracy result for a nonlinear elliptic equation

Let Ω be a smooth bounded domain of with N ≥ 5. In this paper we prove, for ɛ > 0 small, the nondegeneracy of the solution of the problem

Existence and uniqueness for a nonlinear discrete hyperbolic system

The existence, uniqueness and asymptotic behaviour of the solutions to a nonlinear discrete hyperbolic system subject to some extreme conditions and initial data are investigated in a real Hilbert space.     

Revising uniqueness for a nonlinear diffusion-convection equation

This paper provides a generalized and simplified proof of the uniqueness of a weak solution for nonlinear diffusion-convection problems of Stefan type with homogeneous boundary conditions and continuous convection.     

Theorem about existence and uniqueness of continuous solution of nonlocal problem for nonlinear hyperbolic equation

The aim of the paper is to give a theorem about the existence and uniqueness of the continuous solution of a non-linear differential hyperbolic problem with a nonlocal condition in a bounded domain. The Banach theorem about the fixed point is used to prove the existence and uniqueness of the problem considered. The results obtained in this paper can be applied in the theory of elasticity with better effect than the analogous known result with the classical initial condition.     

A uniqueness result for Kirchhoff equations with non-Lipschitz nonlinear term

We consider the second order Cauchy problem

     

Problems with nonlinear boundary conditions for a hyperbolic equation

Two problems with nonlinear boundary conditions are studied. Existence and uniqueness theorems are proved for generalized solutions to each problem.     

Mixed nonlocal problem for a nonlinear singular hyperbolic equation

In this paper, we study a nonlocal mixed problem for a nonlinear hyperbolic equation. Based on some a priori estimates and some density arguments, we prove the well posedness of the associated linear problem. The existence and uniqueness of the weak solution of the nonlinear problem are then established by applying an iterative process based on the obtained results for the linear problem. Copyright © 2009 John Wiley & Sons, Ltd.     

A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem

We study the uniqueness of solution for the following boundary value problem involving a nonlocal equation of Kirchhoff type

     

A general stability result for a nonlinear wave equation with infinite memory

In this work we consider a nonlinear wave problem in the presence of an infinite-memory term and prove an explicit and general stability result. Our approach allows a wider class of kernels, among which those of exponential decay type, usually considered in the literature, are only special cases.     

A uniqueness result for a semilinear parabolic system

We prove that for nonnegative, continuous, bounded and nonzero initial data we have a unique solution of the reaction-diffusion system described by three differential equations with non-Lipschitz nonlinearity. We also find the set of all nonnegative solutions of the system when the initial data is zero and in the last section we briefly discuss a generalization of the theorem to a system of n equations.     

Existence and uniqueness of solutions for a nonlinear fractional differential equation

In this paper we prove the existence and uniqueness of solutions for a class of the nonlinear fractional differential equation with initial condition and investigate the dependence of the solution on the order of the differential equation and on the initial condition. Then we give an example to demonstrate the main results.