44 research outputs found
Existence and Decay of Solutions of a Nonlinear Viscoelastic Problem with a Mixed Nonhomogeneous Condition
We study the initial-boundary value problem for a nonlinear wave equation
given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u)
, 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\eta u(1,t)=g(t),
u(x,0)=\^u_{0}(x), u_{t}(x,0)={\^u}_{1}(x), where \eta \geq 0, q\geq 2 are
given constants {\^u}_{0}, {\^u}_{1}, g, k, f are given functions. In part I
under a certain local Lipschitzian condition on f, a global existence and
uniqueness theorem is proved. The proof is based on the paper [10] associated
to a contraction mapping theorem and standard arguments of density. In Part} 2,
under more restrictive conditions it is proved that the solution u(t) and its
derivative u_{x}(t) decay exponentially to 0 as t tends to infinity.Comment: 26 page
A new stability results for the backward heat equation
In this paper, we regularize the nonlinear inverse time heat problem in the
unbounded region by Fourier method. Some new convergence rates are obtained.
Meanwhile, some quite sharp error estimates between the approximate solution
and exact solution are provided. Especially, the optimal convergence of the
approximate solution at t = 0 is also proved. This work extends to many earlier
results in (f2,f3, hao1,Quan,tau1, tau2, Trong3,x1).Comment: 13 page
The regularity and exponential decay of solution for a linear wave equation associated with two-point boundary conditions
This paper is concerned with the existence and the regularity of global
solutions to the linear wave equation associated with two-point type boundary
conditions. We also investigate the decay properties of the global solutions to
this problem by the construction of a suitable Lyapunov functional.Comment: 18 page
On a nonlinear heat equation associated with Dirichlet -- Robin conditions
This paper is devoted to the study of a nonlinear heat equation associated
with Dirichlet-Robin conditions. At first, we use the Faedo -- Galerkin and the
compactness method to prove existence and uniqueness results. Next, we consider
the properties of solutions. We obtain that if the initial condition is bounded
then so is the solution and we also get asymptotic behavior of solutions as.
Finally, we give numerical resultsComment: 20 page
Existence, blow-up and exponential decay estimates for a nonlinear wave equation with boundary conditions of two-point type
This paper is devoted to study a nonlinear wave equation with boundary
conditions of two-point type. First, we state two local existence theorems and
under suitable conditions, we prove that any weak solutions with negative
initial energy will blow up in finite time. Next, we give a sufficient
condition to guarantee the global existence and exponential decay of weak
solutions. Finally, we present numerical resultsComment: 2
Large time behavior of differential equations with drifted periodic coefficients modeling Carbon storage in soil
This paper is concerned with the linear ODE in the form
, which represents a simplified
storage model of the carbon in the soil. In the first part, we show that, for a
periodic function , a linear drift in the coefficient involves
a linear drift for the solution of this ODE. In the second part, we extend the
previous results to a classical heat non-homogeneous equation. The connection
with an analytic semi-group associated to the ODE equation is considered in the
third part. Numerical examples are given.Comment: 18 page
Determine the source term of a two-dimensional heat equation
Let be a two-dimensional heat conduction body. We consider the
problem of determining the heat source with
be given inexactly and be unknown. The problem is nonlinear and ill-posed.
By a specific form of Fourier transforms, we shall show that the heat source is
determined uniquely by the minimum boundary condition and the temperature
distribution in at the initial time and at the final time .
Using the methods of Tikhonov's regularization and truncated integration, we
construct the regularized solutions. Numerical part is given.Comment: 18 page
Determination of the body force of a two-dimensional isotropic elastic body
Let represent a twodimensional isotropic elastic body. We
consider the problem of determining the body force whose form
with be given inexactly. The problem is
nonlinear and ill-posed. Using the Fourier transform, the methods of Tikhonov's
regularization and truncated integration, we construct a regularized solution
from the data given inexactly and derive the explicitly error estimate.
Numerical part is givenComment: 23 page
Determine the spacial term of a two-dimensional heat source
We consider the problem of determining a pair of functions satisfying
the heat equation , where and the function is given. The problem is
ill-posed. Under a slight condition on , we show that the solution is
determined uniquely from some boundary data and the initial temperature. Using
the interpolation method and the truncated Fourier series, we construct a
regularized solution of the source term from non-smooth data. The error
estimate and numerical experiments are given.Comment: 18 page
Ice formation in the Arctic during summer: false-bottoms
The only source of ice formation in the Arctic during summer is a layer of
ice called false-bottoms between an under-ice melt pond and the underlying
ocean. Of interest is to give a mathematical model in order to determine the
simultaneous growth and ablation of false-bottoms, which is governed by both of
heat fluxes and salt fluxes. In one dimension, this problem may be considered
mathematically as a two-phase Stefan problem with two free boundaries. Our main
result is to prove the existence and uniqueness of the solution from the
initial condition.Comment: 22 page