Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation

We consider the inverse time problem for the nonlinear heat equation in the form

     

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.     

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.     

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.     

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.      

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.     

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.     

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)     

Stability estimates for parabolic problems with Wentzell boundary conditions

Of concern is the uniformly parabolic problem

     

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.     

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.     

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.     

Exact solution of a degenerate fully nonlinear diffusion equation

An exact solution is given for the evolution of an initially v-shaped surface by a fully nonlinear diffusion equation. This is the unique generalized solution that is continuous but not twice differentiable. Since the profile velocity decreases faster than the reciprocal of the profile curvature, the point of infinite curvature persists for a finite positive time.     

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.