Homogeneous Dirichlet Boundary Conditions Research Articles

Existence of solutions for anisotropic parabolic Ni-Serrin type equations originated from a capillary phenomena with nonstandard growth nonlinearity

We consider an initial boundary value problem for a class of anisotropic parabolic Ni-Serrin type equations with nonstandard nonlinearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. Because the nonlinear perturbation leads to difficulties (it does not have a definite sign) in obtaining a priori estimates in the energy method, we had to modify the Tartar method significantly. Under suitable assumptions, we obtain the global existence, decay and extinction of solutions.

On a class of doubly nonlinear evolution equations in Musielak–Orlicz spaces

AbstractThis paper is concerned with a parabolic evolution equation of the form , settled in a smooth bounded domain of , , and complemented with the initial conditions and with (for simplicity) homogeneous Dirichlet boundary conditions. Here, stands for a diffusion operator, possibly nonlinear, which may range in a very wide class, including the Laplacian, the ‐Laplacian for suitable , the “variable‐exponent” ‐Laplacian, or even some fractional order operators. The operator is assumed to be in the form with being measurable in and maximal monotone in . The main results are devoted to proving existence of weak solutions for a wide class of functions that extends the setting considered in previous results related to the variable exponent case where . To this end, a theory of subdifferential operators will be established in Musielak–Orlicz spaces satisfying structure conditions of the so‐called ‐type, and a framework for approximating maximal monotone operators acting in that class of spaces will also be developed. Such a theory is then applied to provide an existence result for a specific equation, but it may have an independent interest in itself. Finally, the existence result is illustrated by presenting a number of specific equations (and, correspondingly, of operators , ) to which the result can be applied.

On the space-time analyticity of the inhomogeneous heat equation on the half space with Neumann boundary conditions

We consider the inhomogeneous heat equation on the half-space R + d with Neumann boundary conditions. We prove a space-time Gevrey regularity of the solution, with a radius of analyticity uniform up to the boundary of the half-space. We also address the case of homogeneous Robin boundary conditions. Our results generalize the case of homogeneous Dirichlet boundary conditions established by Kukavica and Vicol [A direct approach to Gevrey regularity on the half-space. In: Partial differential equations in fluid mechanics. Cambridge: Cambridge Univ. Press; 2018. p. 268–288. (London math. soc. lecture note ser.; vol. 452).].

Unique continuation and observability for a semi-linear parabolic system

This article is concerned with the unique continuation and observability of a semi-linear parabolic system in a bounded domain Ω with  homogeneous Dirichlet boundary conditions. We first study the existence and  uniqueness of the L[[EQUATION]]-solution to the system when the initial value belongs to L[[EQUATION]](Ω)2.  We then build up some observation estimates for this system. As a consequence, we can conclude if two solutions  are equal over  a nonempty open subset ω[[EQUATION]]Ω   at some time T>0, then they coincide over Ω[[EQUATION]][0,T].

The dynamics of an eco-epidemiological prey–predator model with infectious diseases in prey

In this paper, we first propose an eco-epidemiological prey–predator model with infectious diseases in prey. Then study the ODE model, and diffusive model with the homogeneous Neumann and Dirichlet boundary conditions, respectively. For the ODE model and the diffusive model with the homogeneous Neumann boundary conditions, we give a complete conclusion about the stabilities of nonnegative equilibrium states (nonnegative constant equilibrium solutions). The results show that these two problems has no periodic solutions, and the diffusive model with the homogeneous Neumann boundary conditions has no yet Turing patterns. For the diffusive model with the homogeneous Dirichlet boundary conditions, we first establish the necessary and sufficient conditions for the existence of positive equilibrium solutions, and prove that the positive equilibrium solution is unique when it exists. Then we study the global asymptotic stabilities of trivial and semi-trivial nonnegative equilibrium solutions.

Coexistence states in a cross-diffusion system of a competition model

The main goal of this paper is to study the stationary problem for a Lotka–Volterra competition system with advection under homogeneous Dirichlet boundary conditions. By using global bifurcation theory, we establish the sufficient conditions on terms of the birth rates of two competing species assuring the existence of positive solutions. Moreover, some sufficient conditions for the nonexistence of positive solutions are also given. These contrast with the mathematical analyses carried out by Kuto and Tsujikawa (2015) and Wang and Yan (2015), where the corresponding Neumann problem is analyzed.

Long-time behavior for evolution processes associated with non-autonomous nonlinear Schrödinger equation

We consider non-autonomous nonlinear Schrödinger equation with homogeneous Dirichlet boundary conditions in a bounded smooth domain and time-dependent forcing that models the motion of waves in a quantum-mechanical system. We address the problem of the local and global well posedness and using rescaling of time we prove the existence of a compact pullback attractor for the associated evolution process. Moreover, we prove that the pullback attractor has finite fractal dimension. To the best of our knowledge, our approach has not been used in the literature to treat the non-autonomous Schrödinger equation.

Multiple Solutions for a Nonlocal Kirchhoff Problem in Fractional Orlicz-Sobolev Spaces

In this paper, using the three critical points theorem we obtain the existence of three weak solutions for a Kirchhoff type problem driven by a nonlocal operator of the elliptic type in a fractional Orlicz-Sobolev space, with homogeneous Dirichlet boundary conditions.

Simultaneous blow-up of the solution for a singular parabolic system with concentrated sources

<abstract> <p>A singular parabolic system with homogeneous Dirichlet boundary conditions and concentrated nonlinear reaction sources was examined. This paper investigated the existence and uniqueness of the solution. The sufficient conditions for the solution being blown up simultaneously in a finite time were determined. In addition, it was shown that the solution blew up everywhere in the domain except boundary points.</p> </abstract>

Nonlinear elliptic equations by topological degree in Musielak-Orlicz-Sobolev spaces

"We prove by using the topological degree theory the existence of at least one weak solution for the nonlinear elliptic equation − div a1(x, ∇u) + a0(x, u) = f (x, u, ∇u) with homogeneous Dirichlet boundary condition in Musielak-Orlicz-Sobolev spaces. Keywords: Nonlinear elliptic equation, Musielak-Orlicz-Sobolev space, topological degree."

A symmetric low-regularity integrator for the nonlinear Schrödinger equation

Abstract We introduce and analyze a symmetric low-regularity scheme for the nonlinear Schrödinger (NLS) equation beyond classical Fourier-based techniques. We show fractional convergence of the scheme in $L^2$-norm, from first up to second order, both on the torus $\mathbb {T}^d$ and on a smooth bounded domain $\varOmega \subset \mathbb {R}^d$, $d\le 3$, equipped with homogeneous Dirichlet boundary condition. The new scheme allows for a symmetric approximation to the NLS equation in a more general setting than classical splitting, exponential integrators, and low-regularity schemes (i.e., under lower regularity assumptions, on more general domains, and with fractional rates). We motivate and illustrate our findings through numerical experiments, where we witness better structure preserving properties and an improved error-constant in low-regularity regimes.

Determination of Spacewise− Dependent Heat Source Term in Pseudoparabolic Equation from Overdetermination Conditions

      This paper examines the finding of spacewise dependent heat source function in pseudoparabolic equation with initial and homogeneous Dirichlet boundary conditions, as well as the final time value / integral specification as additional conditions that ensure the uniqueness solvability of the inverse problem. However, the problem remains ill-posed because tiny perturbations in input data cause huge errors in outputs. Thus, we employ Tikhonov’s regularization method to restore this instability. In order to choose the best regularization parameter, we employ L-curve method. On the other hand, the direct (forward) problem is solved by a finite difference scheme while the inverse one is reformulated as an optimization problem. The later problem is accomplished by employing lsqnolin subroutine from MATLAB. Two test examples are presented to show the efficiency and accuracy of the employed method by including many noises level and various regularization parameters.

On Lewy Stampacchia inequalities for a pseudomonotone parabolic obstacle problem with [formula omitted]-data

The aim of this paper is to prove the existence of entropy solution, and the associated Lewy-Stampacchia inequalities in a renormalized form, to some parabolic obstacle problems governed by a Leray-Lions pseudomonotone operator in the presence of L1-data and homogeneous Dirichlet boundary conditions.

Large time behavior of signed fractional porous media equations on bounded domains

Following the methodology of Brasco (Adv Math 394:108029, 2022), we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane–Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.

On a multivalued prescribed mean curvature problem and inclusions defined on dual spaces

This article addresses two main objectives. First, it establishes a functional analytic framework and presents existence results for a quasilinear inclusion describing a prescribed mean curvature problem with homogeneous Dirichlet boundary conditions, involving a multivalued lower order term. The formulation of the problem is done in the space of functions with bounded variation. The second objective is to introduce a general existence theory for inclusions defined on nonreflexive Banach spaces, which is specifically applicable to the aforementioned prescribed mean curvature problem. This problem can be formulated as a multivalued variational inequality in the space of functions with bounded variation, which, under suitable conditions, is equivalent to an inclusion involving a maximal monotone mapping of type (D) and a generalized pseudomonotone mapping. We prove an abstract existence theorem for inclusions of this form, under some coercivity conditions involving both the maximal monotone and the generalized pseudomonotone mappings.

Lower Bounds of Blow Up Time for a Class of Slow Reaction Diffusion Equations with Inner Absorption Terms

In this paper, a class of slow reaction-diffusion equations with nonlocal source and inner absorption terms are studied. By using the technique of improved differential inequality, the lower bounds of blow up time for the system under either homogeneous Dirichlet or nonhomogeneous Neumann boundary conditions are obtained.

Global bifurcation of coexistence states for a prey-taxis system with homogeneous Dirichlet boundary conditions

Global bifurcation of coexistence states for a prey-taxis system with homogeneous Dirichlet boundary conditions

Blow‐up and energy decay for a class of wave equations with nonlocal Kirchhoff‐type diffusion and weak damping

The purpose of this paper is to study the following equation driven by a nonlocal integro‐differential operator : with homogeneous Dirichlet boundary condition and initial data, where is the Gagliardo seminorm, and with , is the space dimension. By virtue of a differential inequality technique, an upper bound of the blow‐up time is obtained with a bounded initial energy if and some additional conditions are satisfied. For , in particular, the blow‐up result with high initial energy also is showed by constructing a new control functional and combining energy inequalities with the concavity argument. Moreover, an estimate for the lower bound of the blow‐up time is investigated. Finally, the energy decay estimate is proved as well. These results improve and complement some recent works.

A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration

In recent years, the fractional Laplacian has attracted the attention of many researchers, the corresponding fractional obstacle problems have been widely applied in mathematical finance, particle systems, and elastic theory. Furthermore, the monotonicity of the numerical scheme is beneficial for numerical stability. The purpose of this work is to introduce a monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. Through successful monotone discretization of the fractional Laplacian, the monotonicity is preserved for the fractional obstacle problem and the uniform boundedness, existence, and uniqueness of the numerical solutions of the fractional obstacle problems are proved. A policy iteration is adopted to solve the discrete nonlinear problems, and the convergence after finite iterations can be proved through the monotonicity of the scheme. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the efficacy of the proposed method.

Emergence of time periodic solutions for the generalized surface quasi-geostrophic equation in the disc

In this paper we address the existence of time periodic solutions for the generalized inviscid SQG equation in the unit disc with homogeneous Dirichlet boundary condition when α∈(0,1). We show the existence of a countable family of bifurcating curves from the radial patches. In contrast with the preceding studies in active scalar equations, the Green function is no longer explicit and we circumvent this issue by a suitable splitting into a singular explicit part (which coincides with the planar one) and a smooth implicit one induced by the boundary of the domain. Another problem is connected to the analysis of the linear frequencies which admit a complicated form through a discrete sum involving Bessel functions and their zeros. We overcome this difficulty by using Sneddon's formula leading to a suitable integral representation of the frequencies.