Homogeneous Dirichlet Boundary Conditions Research Articles

On a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity

<p>In this paper, we considered a singular parabolic $ p $-Laplacian equation with logarithmic nonlinearity in a bounded domain with homogeneous Dirichlet boundary conditions. We established the local solvability by the technique of cut-off combining with the method of Faedo-Galerkin approximation. Based on the potential well method and Hardy-Sobolev inequality, the global existence of solutions was derived. In addition, we obtained the results of the decay. The blow-up phenomenon of solutions with different indicator ranges was also given. Moreover, we discussed the blow-up of solutions with arbitrary initial energy and the conditions of extinction.</p>

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.

The sub-supersolution method for variable exponent double phase systems with nonlinear boundary conditions

In this paper, we study quasilinear elliptic systems driven by variable exponent double phase operators involving fully coupled right-hand sides and nonlinear boundary conditions. The aim of our work is to establish an enclosure and existence result for such systems by means of trapping regions formed by pairs of sub- and supersolutions. Under very general assumptions on the data, we then apply our result to get infinitely many solutions. Moreover, we discuss the case when we have homogeneous Dirichlet boundary conditions and present some existence results for this kind of problem.

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.

Probabilistic analysis of a class of 2D-random heat equations via densities

We give new probabilistic results for a class of random two-dimensional homogeneous heat equations with mixed homogeneous Dirichlet and Neumann boundary conditions and an arbitrary initial condition on a rectangular domain. The diffusion coefficient is assumed to be an arbitrary second-order random variable, while the initial condition is a stochastic process admitting a Karhunen-Loève expansion. We then construct pointwise convergent approximations for the main moments and the density of the solution. The theoretical results are numerically illustrated.

Determining Modes of the 2D g-Navier-Stokes Equations

The “determining modes” introduced by Prodi and Foias in 1967 say that if two solutions agree asymptotically in their P projection, then they are asymptotically in their entirety (see (Foias, 1967)). We study the initial boundary value problem for 2D g-Navier-Stokes (g-NVS) equations in bounded domains with homogeneous Dirichlet boundary conditions. We find an improved upper bound on the number of deterministic modes. Moreover, we slightly improve the estimate of the number of deterministic modes and achieve the upper limit of the Grashof Gr numerical order. These estimates are consistent with heuristic estimates based on physical arguments, extends previous results by O.P. Manley and Y.M. Treve (see (Foias, 1983)). The Gronwall lemma and Poincaré type inequality will play a central role in our computational technique as well as of the paper. Studying the properties of solutions is important to determine the behavior of solutions over a long period of time. The obtained result particularly extends previous results for 2D NVS equations.

Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, while the complete segregation can be characterized by the set of nodal solutions of another nonlinear elliptic equation. The second result concerns the global bifurcation structure of the 1d model with large cross-diffusion terms to show that the branch of small coexistence bifurcates from the trivial solution and many branches of complete segregation bifurcate from solutions on the branch of small coexistence. Finally, in order to verify the theoretical results, we chase some bifurcation branches numerically via the continuation software pde2path.

Gradient estimates of general nonlinear singular elliptic equations with measure data

We develop a global Calderón-Zygmund estimate for the gradients of renormalized solutions to the general nonlinear singular elliptic equations −divA(x,u,Du)=μ on a Reifenberg flat domain with the homogeneous Dirichlet boundary condition, while μ is a finite signed Radon measure. The associated nonlinearity behaves as the elliptic p-Laplacian with respect to Du for the singular case p∈(1,2−1/n], whose discontinuity in the x-variable is measured in terms of small BMO, and the Lipschitz continuity is required with respect to the u-variable. We prove it in two folds: the perturbation technique and the weighted Vitali type covering are first employed to establish the weighted good-λ type inequality, then such inequality is used to prove the desired global gradient estimates in weighted Lorentz spaces and Lorentz-Morrey spaces. As a direct consequence, finally we obtain a global gradient regularity in weighted Orlicz spaces.

Quantum-information theory of magnetic field influence on circular dots with different boundary conditions

Influence of the transverse uniform magnetic field B on position (subscript ρ) and momentum (γ) Shannon quantum-information entropies S ρ,γ Fisher informations I ρ,γ and informational energies O ρ,γ is studied theoretically for the 2D circular quantum dots (QDs) whose circumference supports homogeneous either Dirichlet or Neumann boundary condition (BC). Comparative analysis reveals similarities and differences of the influence on the properties of the structure of the surface interaction with the magnetic field. A conspicuous distinction between the two spectra are crossings at the increasing induction of the Neumann energies with the same radial quantum number n and adjacent non-positive angular indices m. At the growing B, either system undergoes Landau condensation when its characteristics turn into their uniform field counterparts. It is shown that for the Dirichlet system this transformation takes place at the smaller magnetic intensities; e.g. the Dirichlet sum on its approach from above to a fundamental limit is at any B smaller than the corresponding Neumann quantity what physically means that the former geometry provides more total information about the position and motion of the particle. It is pointed out that the widely accepted disequilibrium uncertainty relation , with d being a dimensionality of the system, is violated by the Neumann QD in the magnetic field. Comparison with electrostatic harmonic confinement is performed; in particular, contrary to the hard-wall QDs, for this configuration the sums S ρ + S γ and the products I ρ I γ and O ρ O γ do not depend on the field. Physical interpretation is based on the different roles of the two BCs and their interplay with the field: Dirichlet (Neumann) surface is a repulsive (attractive) interface.

Mathematical analysis of an age-since infection and diffusion HIV/AIDS model with treatment adherence and Dirichlet boundary condition

In this paper, an epidemic model with homogeneous Dirichlet boundary condition is formulated to study the joint impact of spatial diffusion, infection age and treatment adherence on the HIV/AIDS transmission among humans. It is an interesting problem to understand the threshold dynamics of HIV/AIDS model with the above three factors. Since the complexity of the model and specificity of the boundary condition, there are two main mathematical challenges: i). the compactness of the solution map is not guaranteed; ii). the explicit expression of the basic reproduction cannot be given even if the parameters are spatially independent. We first discuss the well-posedness of the system, then we identify the basic reproduction number R0 as the spectral radius of the next generation operator, followed by the global attractivity of disease-free steady state when R0<1, the uniform persistence of the disease and the existence of the endemic steady state when R0>1. Numerical simulations are carried out to illustrate our theoretical results, which suggest that the diffusion of individuals has an opposite effect on the disease outbreaks, and improving the treatment compliance of HIV infected individuals can control HIV transmission among the population effectively.