A nonlinear parabolic equation backward in time: Regularization with new error estimates

Consider a nonlinear backward parabolic problem in the form

     

A backward parabolic equation with a time-dependent coefficient: Regularization and error estimates

We consider the problem of determining the temperature u(x,t)$u(x,t)$, for (x,t)∈[0,π]×[0,T)$(x,t)\in [0,\pi ]\times [0,T)$ in the parabolic equation with a time-dependent coefficient. This problem is severely ill-posed, i.e., the solution (if it exists) does not depend continuously on the given data. In this paper, we use a modified method for regularizing the problem and derive an optimal stability estimation. A numerical experiment is presented for illustrating the estimate.     

Sharp estimates for approximations to a nonlinear backward heat equation

A nonlinear backward heat problem for an infinite strip is considered. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we use the Fourier regularization method to solve the problem. Some sharp estimates of the error between the exact solution and its regularization approximation are given.     

On a backward Cauchy problem associated with continuous spectrum operator

The nonlinear backward Cauchy problem

u_t+Au(t)=f(u(t)),u(T)=φ,     

The heat equation with nonlinear generalized Robin boundary conditions

Let Ω⊂RN be a bounded domain and let μ be an admissible measure on ∂Ω. We show in the first part that if Ω has the H¹-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by on ∂Ω, generates a strongly continuous nonlinear submarkovian semigroup SB=(SB(t))_t?0 on L²(Ω). We also obtain that this semigroup is ultracontractive in the sense that for every u,v∈Lp(Ω), p?2 and every t>0, one has

     

A backward nonlinear heat equation: regularization with error estimates

In this article the authors consider a backward nonlinear heat equation. The uniqueness of the problem is proved and the problem is regularized by finite dimensional approximations. Error estimates in some particular cases are given.     

Existence for a degenerate diffusion problem with a nonlinear operator

We study a degenerate nonlinear variational inequality which can be reduced to a multivalued inclusion by an appropriate change of the unknown function. We establish existence, uniqueness and regularity results. An application arising in the theory of water diffusion in porous media is discussed as an example.      

Solution of nonlinear integral equations of Hammerstein type

Let E be a 2-uniformly real Banach space and F,K:E→E be nonlinear-bounded accretive operators. Assume that the Hammerstein equation u+KFu=0 has a solution. A new explicit iteration sequence is introduced and strong convergence of the sequence to a solution of the Hammerstein equation is proved. The operators F and K are not required to satisfy the so-called range condition. No invertibility assumption is imposed on the operator K and F is not restricted to be an angle-bounded (necessarily linear) operator.     

The Hukuhara–Kneser property for a nonlinear integral equation

Applying a structure theorem of Krasnosel’skii and Perov, we show that the solution set of a nonlinear integral equation satisfies the classical Hukuhara–Kneser property.     

Strong convergence theorems for a finite family of nonexpansive mappings and semigroups via the hybrid method

Strong convergence theorems are obtained for a finite family of nonexpansive mappings and semigroups by the hybrid method.     

Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy

In this paper we investigate the asymptotic behavior of the nonlinear Cahn–Hilliard equation with a logarithmic free energy and similar singular free energies. We prove an existence and uniqueness result with the help of monotone operator methods, which differs from the known proofs based on approximation by smooth potentials. Moreover, we apply the Lojasiewicz–Simon inequality to show that each solution converges to a steady state as time tends to infinity.     

Uniqueness for an elliptic-parabolic problem with Neumann boundary condition

We consider the problem in a smooth boundary domain , as well as the corresponding evolution equation . For the stationary equation we show existence results, then we adapt the techniques of doubling of variables to the case of the homogeneous Neumann boundary conditions and obtain the appropriate L ¹ -contraction principle and uniqueness. Subsequently, we are able to apply the nonlinear semigroup theory and prove the L ¹ -contraction principle for the associated evolution equation.     

Explicit relation between the solutions of the heat and the Hermite heat equation

There are lots of results on the solutions of the heat equation but much less on those of the Hermite heat equation due to that its coefficients are not constant and even not bounded. In this paper, we find an explicit relation between the solutions of these two equations, thus all known results on the heat equation can be transferred to results on the Hermite heat equation, which should be a completely new idea to study the Hermite equation. Some examples are given to show that known results on the Hermite equation are obtained easily by this method, even improved. There is also a new uniqueness theorem with a very general condition for the Hermite equation, which answers a question in a paper in Proc. Japan Acad. (2005). Supported partially by 973 project (2004CB318000)     

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.     

Strong convergence theorems for a finite family of asymptotically nonexpansive mappings and semigroups

Strong convergence theorems are obtained for a finite family of asymptotically nonexpansive mappings and semigroups by the modified Mann method.     

Intrinsic ultracontractivity of Dirichlet Laplacians in non-smooth domains

We give a general criterion for the intrinsic ultracontractivity of Dirichlet Laplacians – _D on domainsD ofR ^d d 3, based on the Lieb's formula. It applies to various classes of domains (e.g. John, Hölder andL ^p-averaging domains) and gives new conditions for intrinsic ultracontractivity in terms of the Minkowski dimension of the boundary D. In particular, isotropic self-similar fractals and domains satisfying a c-covering condition are considered.     

Integrable solutions of a nonlinear functional integral equation on an unbounded interval

In this paper we prove the existence of integrable solutions of a generalized functional-integral equation, which includes many key integral and functional equations that arise in nonlinear analysis and its applications. This is achieved by means of an improvement of a Krasnosel’skii type fixed point theorem recently proved by K. Latrach and the author. The result presented in this paper extends the corresponding result of [J. Banas, A. Chlebowicz, On existence of integrable solutions of a functional integral equation under Carathéodory condition, Nonlinear Anal. (2008) doi:10.1016/j.na.2008.04.020]. An example which shows the importance and the applicability of our result is also included.     

Convergence analysis of a Halpern type algorithm for accretive operators

In this paper, we study the convergence of a Halpern type proximal point algorithm for accretive operators in Banach spaces. Our results fill the gap in the work of Zhang and Song (2012) [1] and, consequently, all the results there can be corrected accordingly.     

Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces

In this paper, a general system of nonlinear variational inequality problem in Banach spaces was considered, which includes some existing problems as special cases. For solving this nonlinear variational inequality problem, we construct two methods which were inspired and motivated by Korpelevich’s extragradient method. Furthermore, we prove that the suggested algorithms converge strongly to some solutions of the studied variational inequality.