Related papers: On the nonlinear Dirac equation on noncompact metr…
In this paper, we study local well-posedness and orbital stability of standing waves for a singularly perturbed one-dimensional nonlinear Klein-Gordon equation. We first establish local well-posedness of the Cauchy problem by a fixed point…
In this paper we study the Cauchy problem for second order strictly hyperbolic operators when the coefficients of the principal part are not Lipschitz continuous, but only "Log-Lipschitz" with respect to all the variables. This class of…
We study the well-posedness of the Cauchy problem with Dirichlet or Neumann boundary conditions associated to an H 1 -critical semilinear wave equation on a smooth bounded 2D domain {\Omega}. First, we prove an appropriate Strichartz type…
We study the positive, regular, radially symmetric solutions to the nonlinear biharmonic equation $\Delta^2 \phi = \phi^p$. First, we show that there exists a critical value $p_c$, depending on the space dimension, such that the solutions…
In this paper, we study the existence and multiplicity of solutions to the following class of nonlinear Dirac equations (NLDE) on noncompact quantum graphs: \[ -i\,\varepsilon c\,\sigma_1\,\partial_x u + m c^2 \sigma_3 u + V(x)\,u =…
Problem for the first order differential equation with an unbounded operator coefficient in Banach space and nonlinear nonlocal condition is considered. A numerical method is proposed and justified for the solution of this problem under…
Exact solution of Dirac equation for a particle whose potential energy and mass are inversely proportional to the distance from the force centre has been found. The bound states exist provided the length scale $a$ which appears in the…
We consider a degenerate wave equation in one dimension, with drift and in presence of a leading operator which is not in divergence form. We impose a homogeneous Dirichlet boundary condition where the degeneracy occurs and a boundary…
We consider the one-dimensional Gross-Pitaevskii equation perturbed by a Dirac potential. Using a fine analysis of the properties of the linear propagator, we study the well-posedness of the Cauchy Problem in the energy space of functions…
This work studies the dynamics of solutions to the sine-Gordon equation posed on a tadpole graph $G$ and endowed with boundary conditions at the vertex of $\delta$-type. The latter generalize conditions of Neumann-Kirchhoff type. The…
We deal with the as yet unresolved exponential stability problem for Beck's Problem on a metric star graph with three identical edges. The edges are stretched Euler--Bernoulli beams which are simply supported with respect to the outer…
In this work, we study the higher order Kirchhoff type Choquard equation $(KC)$ involving a critical exponential non-linearity and singular weights. We prove the existence of solution to $(KC)$ using Mountain pass Lemma in light of…
We consider the Cauchy problem defined for a general class of nonlocal wave equations modeling bidirectional wave propagation in a nonlocally and nonlinearly elastic medium whose constitutive equation is given by a convolution integral. We…
In this paper we study existence and uniqueness of solutions for a very general class of doubly nonlinear diffusion equations on metric graphs, which provide the appropriate mathematical framework to describe complex tubular networks in…
We consider the dynamics of relativistic spin-half particles in quantum graphs with transparent branching points. The system is modeled by combining the quantum graph concept with the one of transparent boundary conditions applied to the…
We study the Korteweg--de Vries equation on a metric star graph and investigate existence of solitary waves on the metric graph in terms of the coefficients of the equation on each edge, the coupling condition at the central vertex of the…
We investigate the local energy decay of solutions of the Dirac equation in the non-extreme Kerr-Newman metric. First, we write the Dirac equation as a Cauchy problem and define the Dirac operator. It is shown that the Dirac operator is…
In this paper, we study the schrodinger equation and wave equation with the Dirichlet boundary condition on a connected finite graph. The explicit expressions for solutions are given and the energy conservations are derived. Applications to…
We study low regularity behavior of the nonlinear wave equation in $\mathbb{R}^2$ augmented by the viscous dissipative effects described by the Dirichlet-Neumann operator. Problems of this type arise in fluid-structure interaction where the…
Initial value problems for the integrable discrete equations on quad-graphs are investigated. A geometric criterion of the well-posedness of such a problem is found. The effects of the interaction of the solutions with the localized defects…