Semilinear Problems Research Articles

On a semilinear parabolic problem with non-local (Bitsadze–Samarskii type) boundary conditions in more dimensions

This paper studies a semilinear parabolic equation in a bounded domain Ω⊂Rd along with nonlocal boundary conditions. The boundary values are linked to the values of a solution on an interior (d−1)-dimensional manifold lying inside Ω. Firstly, the solvability of a steady-state problem is addressed. Secondly, involving the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem in a weighted Hilbert space is shown. Convergence of approximations is addressed and the error estimated are derived. Numerical experiments support the theoretical algorithms.

Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion

In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from ones used in the previous published literature on the long time behavior of heat equations with memory, which is carried out by the Dafermos transformation. As a consequence, the obtained results provide complete information about the attracting sets for the original problem, instead of the transformed one. In particular, the proved results also generalize and complete previous literature in the local case.

A Nonlinear Fractional Problem with a Second Kind Integral Condition for Time-Fractional Partial Differential Equation

The aim of this research is to demonstrate the existence and the uniqueness of the weak solution for a semilinear fractional parabolic problem with the special case of the second integral boundary condition. For this aim, we split the proof into two parts; to study the main linear problem part, we used the variable separation method, and concerning the semilinear problem part, we apply an iterative method and a priori estimate for the study of the weak solution.

A NSFD Discretization of Two-Dimensional Singularly Perturbed Semilinear Convection-Diffusion Problems

Despite the availability of an abundant literature on singularly perturbed problems, interest toward non-linear problems has been limited. In particular, parameter-uniform methods for singularly perturbed semilinear problems are quasi-non-existent. In this article, we study a two-dimensional semilinear singularly perturbed convection-diffusion problems. Our approach requires linearization of the continuous semilinear problem using the quasilinearization technique. We then discretize the resulting linear problems in the framework of non-standard finite difference methods. A rigorous convergence analysis is conducted showing that the proposed method is first-order parameter-uniform convergent. Finally, two test examples are used to validate the theoretical findings.

Interface regularity for semilinear one-phase problems

We study critical points of a one-parameter family of functionals arising in combustion models. The problems we consider converge, for infinitesimal values of the parameter, to Bernoulli's free boundary problem, also known as one-phase problem. We prove a C1,α estimates for the “interfaces” (level sets separating the burnt and unburnt regions). As a byproduct, we obtain the one-dimensional symmetry of minimizers in the whole RN, for N≤4, answering positively a conjecture of Fernández-Real and Ros-Oton.Our results are to Bernoulli's free boundary problem what Savin's results for the Allen-Cahn equation are to minimal surfaces.

Nonuniqueness of minimizers for semilinear optimal control problems

A counterexample to uniqueness of global minimizers of semilinear optimal control problems is given. The lack of uniqueness occurs for a special choice of the state-target in the cost functional. Our arguments also show that, for some state-targets, there exist local minimizers which are not global. When this occurs, gradient-type algorithms may be trapped by local minimizers, thus missing global ones. Furthermore, the issue of convexity of a quadratic functional in optimal control is analyzed in an abstract setting. As a corollary of nonuniqueness of minimizers, a nonuniqueness result for a coupled elliptic system is deduced. Numerical simulations have been performed illustrating the theoretical results. We also discuss the possible impact of the multiplicity of minimizers on the turnpike property in long time horizons.

On the Neumann problem for fractional semilinear elliptic equations arising from Keller–Segel model

In this paper, we study the following fractional semilinear Neumann problem arising from Keller–Segel model: where is a smooth bounded domain and is the outward unit normal to . First, under the superlinear and subcritical growth assumptions on , we prove that there exists at least one positive nonconstant solution for small and the family of solutions is uniformly bounded. Moreover, we build a Pohozaev‐type identity for the ( ‐) Neumann harmonic extension of the following problem: where is a function such that . As a direct application of this identity, when satisfies , where , we deduce the nonexistence of weak bounded solution in star‐shaped domains.

Strong solutions for semilinear problems with almost sectorial operators

Strong solutions for semilinear problems with almost sectorial operators

(ω,c)-Periodic Solutions to Fractional Differential Equations with Impulses

This paper deals with the (ω,c)-periodic solutions to impulsive fractional differential equations with Caputo fractional derivative with a fixed lower limit. Firstly, a necessary and sufficient condition of the existence of (ω,c)-periodic solutions to linear problem is given. Secondly, the existence and uniqueness of (ω,c)-periodic solutions to semilinear problem are proven. Lastly, two examples are given to demonstrate our results.

Immersed finite element approximation for semi-linear parabolic interface problems combining with two-grid methods

In this paper, we propose and analyze the two-grid immersed finite element methods for semi-linear parabolic interface problems with discontinuous diffusion coefficients. The immersed finite element methods are used for spatial discretization where the meshes are not aligned with the interface. Optimal error estimates have been derived for both spatially semi-discrete schemes and fully discrete schemes. The two-grid algorithms based on the Newton methods are adopted to treat the nonlinear term. It is theoretically and numerically illustrated that the two-grid immersed finite element methods can achieve optimal convergence order when the coarse mesh satisfies H=O(h1/2) (or H=O(h1/4)).

Weak formulations of the nonlinear Poisson-Boltzmann equation in biomolecular electrostatics

We consider the nonlinear Poisson-Boltzmann equation in the context of electrostatic models for a biological macromolecule, embedded in a bounded domain containing a solution of an arbitrary number of ionic species which is not necessarily charge neutral. The resulting semilinear elliptic equation combines several difficulties: exponential growth and lack of sign preservation in the nonlinearity accounting for ion mobility, measure data arising from point charges inside the molecule, and discontinuous permittivities across the molecule boundary. Exploiting the modeling assumption that the point sources and the nonlinearity are active on disjoint parts of the domain, one can use a linear decomposition of the potential into regular and singular components. A variational argument can be used for the regular part, but the unbounded nonlinearity makes the corresponding functional not differentiable in Sobolev spaces. By proving boundedness of minimizers, these are related to standard H1 weak formulations for the regular component and in the framework of Boccardo and Gallouët for the full potential. Finally, a result of uniqueness of this type of weak solutions for more general semilinear problems with measure data validates the strategy, since the different decompositions and test spaces considered must then lead to the same solution.

Smoothing effect and asymptotic dynamics of nonautonomous parabolic equations with time-dependent linear operators

In this paper we consider the nonautonomous semilinear parabolic problems with time-dependent linear operatorsut+A(t)u=f(t,u), t>τ;u(τ)=u0, in a Banach space X. Under suitable conditions, we obtain regularity results for ut(t,x) with respect to its spatial variable x and estimates for ut in stronger spaces (Xα). We then apply those results to a nonautonomous reaction-diffusion equationut−div(a(t,x)∇u)+u=f(t,u) with Neumann boundary condition and time-dependent diffusion. From the regularity of ut we derive the existence of classical solutions and from the estimates for ut we prove that the variation of the solution u is bounded in the long-time dynamics. We also prove the existence of pullback attractor, as well as the existence of a compact set that contains the long-time dynamics of the derivatives ut, without requiring any assumption concerning monotonicity or decay in time of a(t,x).

A new symmetric mixed element method for semi-linear parabolic problem based on two-grid discretization

Based on two-grid discretization, a new numerical method is constructed for semilinear parabolic problem. In this method, a new symmetric positive definite numerical scheme is provided, and then the optimal a priori error estimates in L2 and Lp-norm are given. In order to improve the efficiency of computation, the two-grid method is considered. The corresponding convergence analysis is studied, and the optimal error estimate is derived under the relation h=O(H2). Some numerical experiments are presented to confirm our theoretical analysis.

The domain derivative for semilinear elliptic inverse obstacle problems

<p style='text-indent:20px;'>We consider the recovering of the shape of a cavity from the Cauchy datum on an accessible boundary in case of semilinear boundary value problems. Existence and a characterization of the domain derivative of solutions of semilinear elliptic equations are proven. Furthermore, the result is applied to solve an inverse obstacle problem with an iterative regularization scheme. By some numerical examples its performance in case of a Kerr type nonlinearity is illustrated.</p>

New results concerning the hierarchical control of linear and semilinear parabolic equations

This paper deals with the application of multiple strategies to control some parabolic PDEs. We assume that we can act on the system through a hierarchy of distributed controls: with a first control (a follower), we drive the state exactly to zero; then, with an additional control (the leader), we minimize a prescribed cost functional. That means that we invert the roles played by leaders and followers in the recent literature. We study linear and semilinear problems. More precisely, we prove the existence (and uniqueness in the linear case) of a leader-follower couple. Then, we deduce an appropriate optimality system that must be satisfied by the controls and the corresponding state and adjoint states. We also indicate some generalizations to other controls, PDEs and systems. In particular, we establish similar existence and optimality results for hierarchical-biobjective (Pareto-Stackelberg) control problems, where there are two cost functionals and two independent leader controls whose main task is to find an associated Pareto equilibrium and one common follower in charge of null controllability.

A semilinear problem with a gradient term in the nonlinearity

<p style='text-indent:20px;'>We consider the following semilinear problem with a gradient term in the nonlinearity</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{align*} -\Delta u = \lambda \frac{(1+|\nabla u|^q)}{(1-u)^p}\quad\text{in}\quad\Omega,\quad u>0\quad \text{in}\quad \Omega, \quad u = 0\quad\text{on}\quad \partial \Omega. \end{align*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda,p,q>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> be a bounded, smooth domain in <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb R}^N $\end{document}</tex-math></inline-formula>. We prove that when <inline-formula><tex-math id="M4">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a unit ball and <inline-formula><tex-math id="M5">\begin{document}$ p = 1 $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M6">\begin{document}$ q\in (0,q^*(N)) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ q^*(N)\in (1,2) $\end{document}</tex-math></inline-formula>, we have infinitely many radial solutions for <inline-formula><tex-math id="M8">\begin{document}$ 2\leq N<2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \lambda = \tilde \lambda $\end{document}</tex-math></inline-formula>. On the other hand, for <inline-formula><tex-math id="M10">\begin{document}$ N>2\frac{6-q+2\sqrt{8-2q}}{(2-q)^2}+1 $\end{document}</tex-math></inline-formula> there exists a unique radial solution for <inline-formula><tex-math id="M11">\begin{document}$ 0<\lambda<\tilde \lambda $\end{document}</tex-math></inline-formula>.</p>

The well-posedness for semilinear time fractional wave equations on $ \mathbb R^N $

<abstract><p>This paper is concerned with the semilinear time fractional wave equations on the whole Euclidean space, also known as the super-diffusive equations. Considering the initial data in the fractional Sobolev spaces, we prove the local/global well-posedness results of $ L^2 $-solutions for linear and semilinear problems. The methods of this paper rely upon the relevant wave operators estimates, Sobolev embedding and fixed point arguments.</p></abstract>

Fully Discrete Interpolation Coefficients Mixed Finite Element Methods for Semi-Linear Parabolic Optimal Control Problem

We use the fully discrete interpolation coefficient mixed finite element methods to solve the semi-linear parabolic optimal control problems. The space discretization of the state variable is separated using interpolation coefficient mixed finite elements. We approximate the state and the co-state by Raviart- Thomas mixed finite elements on the lowest order, which is approximated by piecewise constant elements. By applying mixed finite element methods to estimate errors, we get a priori error estimate for both the coupled state and the control approximations. We finally confirm the theoretical results numerically by a numerical example.

Boundary layer solutions to singularly perturbed quasilinear systems

<p style='text-indent:20px;'>We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon^2\left(A(x, u(x), \varepsilon)u'(x)\right)' = f(x, u(x), \varepsilon) $\end{document}</tex-math></inline-formula>. The new features are that we do not consider one scalar equation, but systems, that the systems are allowed to be quasilinear, and that the systems are spatially non-smooth. Although the results about existence, asymptotic behavior, local uniqueness and stability of boundary layer solutions are similar to those known for semilinear, scalar and smooth problems, there are at least three essential differences. First, the asymptotic convergence rates valid for smooth problems are not true anymore, in general, in the non-smooth case. Second, a specific local uniqueness condition from the scalar case is not sufficient anymore in the vectorial case. And third, the monotonicity condition, which is sufficient for stability of boundary layers in the scalar case, must be adjusted to the vectorial case.</p>

Existence and uniqueness of solutions to some classes of nonlocal semilinear conformable fractional differential or integrodifferential equations

The purpose of this paper is to give and prove the fundamental theorem of conformable fractional calculus which is given only in previous associated papers for differentiable functions, and then we scrutinize the existence and uniqueness of solutions to some semilinear Cauchy problems for nonlocal conformable fractional integrodifferential or differential equations which their nonlinear terms include fractional derivative or fractional integral. Two examples are investigated to elucidate the main results.