Homogeneous Boundary Research Articles

Blow-up and non-extinction for a nonlocal parabolic equation with logarithmic nonlinearity

This paper is devoted to studying a nonlocal parabolic equation with logarithmic nonlinearity ulog |u|-fint_{Omega } ulog |u|,dx in a bounded domain, subject to homogeneous Neumann boundary value condition. By using the logarithmic Sobolev inequality and energy estimate methods, we get the results under appropriate conditions on blow-up and non-extinction of the solutions, which extend some recent results.

A quasilinear parabolic–elliptic chemotaxis-growth system with nonlinear secretion

ABSTRACTThis paper deals with the parabolic–elliptic chemotaxis-growth system with nonlinear secretion under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary, where , a>0, b>0, r>1, and is a smooth function satisfying with some and. It is shown, either or or then the system admits a unique global bounded classical solution. Moreover, we obtain the large time behavior of the solution for a specific logistic source. Finally, for the case and r=k+1, large time behavior and convergence rate are established.

Nonlinear mixed Cherepanov boundary-value problem

ABSTRACTWe consider a nonlinear boundary-value problem consisting in the determination of a function that is meromorphic in the upper half-plane, satisfies the homogeneous Hilbert boundary condition on the set L of n intervals of the real axis and has a given module on the set . This problem was stated and solved in Cherepanov. Cherepanov proved that the required solution with a given number and location of its interior zeros and poles and with integrable singularities at all endpoints of L exists if and only if n−1 solvability conditions are satisfied. We prove that this problem is unconditionally solvable. A particular solution, , is found in the class of meromorphic functions with a properly chosen location of n−1 zeros and poles. Namely, we have shown that these zeros and poles are defined as the solution of some solvable real analogue of the Jacobi inversion problem. A general meromorphic solution of the Cherepanov problem is obtained with the help of the particular solution . The problem of a possible decrease in the number of zeros and poles of the desired solution is investigated.

Asymptotic analysis of a multiscale parabolic problem with a rough fast oscillating interface

This paper is concerned with the well posedness and homogenization for a multiscale parabolic problem in a cylinder Q of $${\mathbb {R}}^N $$ . A rapidly oscillating non-smooth interface inside Q separates the cylinder in two heterogeneous connected components. The interface has a periodic microstructure, and it is situated in a small neighborhood of a hyperplane which separates the two components of Q. The problem models a time-dependent heat transfer in two heterogeneous conducting materials with an imperfect contact between them. At the interface, we suppose that the flux is continuous and that the jump of the solution is proportional to the flux. On the exterior boundary, homogeneous Dirichlet boundary conditions are prescribed. We also derive a corrector result showing the accuracy of our approximation in the energy norm.

Electrical and optical properties of lead-free 0.15(K0.5Bi0.5TiO3)–0.85(NaNbO3) solid solution

Polycrystalline lead free 0.15(K0.5Bi0.5TiO3)–0.85(NaNbO3) ceramic is synthesized using conventional solid state reaction technique. Structural, dielectric, optical and impedance properties of the sample are investigated. X-ray diffraction (XRD) study reveals K0.5B0.5TiO3 (KBT) diffusing into the NaNbO3 (NN) lattices to form a new perovskite-type solid solution with orthorhombic structure (space group Pmc2 1 ) at room temperature. Addition of small amount of KBT transforms NN ceramic from anti-ferroelectric to ferroelectric material. The XRD data has been refined using Rietveld refinement technique. FTIR spectra reflect an appreciable change in its vibrational phonon modes of the sample. The optical band gap is estimated to be 3.02 eV from the diffused absorbance spectra, which is useful in photo catalytic applications. Different vibrational modes in the frequency range of 130–900 cm−1 is observed from the Raman spectrum. FESEM micrograph reveals the homogeneous grains and well-defined grain boundaries with some few pores. Dielectric and complex impedance spectroscopic studies are carried out in a wide range of frequency (i.e., 103–106 Hz) and temperature (30–475 °C). The presence of ferroelectric properties is confirmed from P–E loop. A strong correlation between its microstructure and electrical parameters is established from the study of the electrical properties of the material. The ac conductivity spectrum obeys Jonscher’s power law. Negative temperature coefficient of resistance behavior of the material is inferred from temperature dependence of dc conductivity.

A note on the existence of time periodic solution of a superlinear heat equation

In this paper, we consider the existence of positive periodic solutions for a superlinear heat equation with and being the Sobolev critical exponent, subject to homogeneous Dirichlet boundary condition. We relax the technical assumption on a(t) and convex assumption on the domain , which generalizes the previous results.

Analytical Solution for 2D Inter-Well Porous Flow in a Rectangular Reservoir

Inter-well fluid flows through porous media are commonly encountered in the production of groundwater, oil, and geothermal energy. In this paper, inter-well porous flow inside a rectangular reservoir is solved based on the complex variable function theory combined with the method of mirror images. In order to derive the solution analytically, the inter-well flow is modeled as a 2D flow in a homogenous and isotropic porous medium. The resulted exact analytical solution takes the form of an infinite series, but it can be truncated to give high accuracy approximation. In terms of nine cases of inter-well porous flow associated with enhanced geothermal systems, the applications of the obtained analytical solution are demonstrated, and the convergence properties of the truncated series are investigated. It is shown that the convergent rate of the truncated series increases with the symmetric level of well distribution inside the reservoir, and the adoption of Euler transform significantly accelerates the convergence of alternating series cases associated with asymmetric well distribution. In principle, the analytical solution proposed in this paper can be applied to other scientific and engineering fields, as long as the involved problem is governed by 2D Laplace equation in a rectangular domain and subject to similar source/sink and boundary conditions, i.e., isolated point sources/sinks and uniform Dirichlet or homogeneous Neumann boundary conditions.

The Kuramoto–Sivashinsky equation. A Local Attractor Filled with Unstable Periodic Solutions

A periodic boundary value problem is considered for one version of the KuramotoSivashinsky equation, which is widely known in mathematical physics. Local bifurcations in a neighborhood of the spatially homogeneous equilibrium points in the case when they change stability are studied. It is shown that the loss of stability of homogeneous equilibrium points leads to the appearance of a two-dimensional attractor on which all solutions are periodic functions of time, except one spatially inhomogeneous state. A spectrum of frequencies of the given family of periodic solutions fills the entire number line, and they are all unstable in a sense of Lyapunov definition in the metric of the phase space (space of initial conditions) of the corresponding initial boundary value problem. It is chosen the Sobolev space as the phase space. For the periodic solutions which fill the two-dimensional attractor, the asymptotic formulas are given. In order to analyze the bifurcation problem it was used analysis methods for infinite-dimensional dynamical systems: the integral (invariant) manifold method, the Poincare normal form theory, and asymptotic methods. The analysis of bifurcations for periodic boundary value problem was reduced to analysing the structure of the neighborhood of the zero solution of the homogeneous Dirichlet boundary value problem for the considered equation.

Two-Level Block Decompositions for Solving Helmholtz Equation via Chebyshev Pseudo Spectral Method

In this paper, we consider solving the Helmholtz equation in the Cartesian domain , subject to homogeneous Dirichlet boundary condition, discretized with the Chebyshev pseudo-spectral method. The main purpose of this paper is to present the formulation of a two-level decomposition scheme for decoupling the linear system obtained from the discretization into independent subsystems. This scheme takes advantage of the homogeneity property of the physical problem along one direction to reduce a 2D problem to several 1D problems via a block diagonalization approach and the reflexivity property along the second direction to decompose each of the 1D problems to two independent subproblems using a reflexive decomposition, effectively doubling the number of subproblems. Based on the special structure of the coefficient matrix of the linear system derived from the discretization and a reflexivity property of the second-order Chebyshev differentiation matrix, we show that the decomposed submatrices exhibits a similar property, enabling the system to be decomposed using reflexive decompositions. Explicit forms of the decomposed submatrices are derived. The decomposition not only yields more efficient algorithm but introduces coarse-grain parallelism. Furthermore, it preserves all eigenvalues of the original matrix.

Effective Methods for Solving Band SLEs after Parabolic Nonlinear PDEs

A class of models of heat transfer processes in a multilayer domain is considered. The governing equation is a nonlinear heat-transfer equation with different temperature-dependent densities and thermal coefficients in each layer. Homogeneous Neumann boundary conditions and ideal contact ones are applied. A finite difference scheme on a special uneven mesh with a second-order approximation in the case of a piecewise constant spatial step is built. This discretization leads to a pentadiagonal system of linear equations (SLEs) with a matrix which is neither diagonally dominant, nor positive definite. Two different methods for solving such a SLE are developed - diagonal dominantization and symbolic algorithms.

A note on the homogeneous Neumann boundary contidion

A note on the homogeneous Neumann boundary contidion

Dynamics for the damped wave equations on time-dependent domains

We consider the asymptotic dynamics of a damped wave equations on a time-dependent domains with homogeneous Dirichlet boundary condition, the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. To this end, we establish the existence and uniqueness of strong and weak solutions satisfying energy inequality under the assumption that the spatial domains $\mathcal{O}_{t}$ in $\mathbb{R}^{3}$ are obtained from a bounded base domain $\mathcal{O}$ by a $C^{3}$-diffeomorphism $r(·, t)$. Furthermore, we establish the pullback attractor under a slightly weaker assumption that the measure of the spatial domains are uniformly bounded above.

Hölder Estimates for the Regular Component of the Solution to a Singularly Perturbed Convection–Diffusion Equation

In a half-plane, a homogeneous Dirichlet boundary value problem for an inhomogeneous singularly perturbed convection–diffusion equation with constant coefficients and convection directed orthogonally away from the boundary of the half-plane is considered. Assuming that the right-hand side of the equation belongs to the space Cλ, 0 < λ < 1, and the solution is bounded at infinity, an unimprovable estimate of the solution is obtained in a corresponding Holder norm (anisotropic with respect to a small parameter).

Boundedness of classical solutions for a chemotaxis system with general sensitivity function

ABSTRACTThis paper deals with the chemotaxis system with general sensitivity function: under homogeneous Neumann boundary conditions in a bounded domain with smooth boundary. Here, and the initial function and the sensitivity function satisfy: We prove that the classical solutions to the above system are uniformly in-time-bounded provided that there exists a smooth positive function such that for some and the following differential inequality holds where and is bounded from above. We also present our results for the special case with and These results coincide with the results obtained by Fujie et al. [Math Methods Appl Sci. 2015] in the case of and extend their results in the case of .

Existence and regularity of solutions to a quasilinear elliptic problem involving variable sources

The authors of this paper prove the existence and regularity results for the homogeneous Dirichlet boundary value problem to the equation -operatorname{div}(|nabla u|^{p-2}nabla u)=frac{f(x)}{u ^{alpha(x)}} with fin L^{m}(Omega) (mgeqslant1) and alpha(x)>0. Due to the nonlinearity of a p-Laplace operator and the anisotropic variable exponent alpha(x), some classical methods may not directly be applied to our problem. In this paper, we construct a suitable test function and apply the Leray-Schauder fixed point theorem to prove the existence of positive solutions with necessary a priori estimate and compact argument. Furthermore, we also discuss the relationship among the regularity of solutions, the summability of f and the value of alpha(x).

Existence and uniqueness for some nonlocal elliptic problem

In this paper, we consider the existence and uniqueness of solutions for the nonlocal elliptic problem with homogeneous Dirichlet boundary conditions, where , . It will be shown that: (i) when , there exists a unique solution to the nonlocal elliptic problem for any ; (ii) when , there also exists a unique solution of the nonlocal elliptic problem for and no solution for .

Quantitative unique continuation for the linear coupled heat equations

In this paper, we established a quantitative unique continuation results for a coupled heat equations, with the homogeneous Dirichlet boundary condition, on a bounded convex domain Ω of mathbb{R}^{d} with smooth boundary ∂Ω. Our result shows that the value of the solutions can be determined uniquely by its value on an arbitrary open subset ω of Ω at any given positive time T.

Positive steady states of a density-dependent predator-prey model with diffusion

In this paper, we investigate the rich dynamics of a diffusive Holling type-Ⅱ predator-prey model with density-dependent death rate for the predator under homogeneous Neumann boundary condition. The value of this study lies in two-aspects. Mathematically, we show the stability of the constant positive steady state solution, the existence and nonexistence, the local and global structure of nonconstant positive steady state solutions. And biologically, we find that Turing instability is induced by the density-dependent death rate, and both the general stationary pattern and Turing pattern can be observed as a result of diffusion.

Isospectral Domains for Discrete Elliptic Operators

Concerning the Laplace operator with homogeneous Dirichlet boundary conditions, the classical notion of isospectrality assumes that two domains are related when they give rise to the same spectrum. In two dimensions, non isometric, isospectral domains exist. It is not known however if all the eigenvalues relative to a specific domain can be preserved under suitable continuous deformation of its geometry. We show that this is possible when the 2D Laplacian is replaced by a finite dimensional version and the geometry is modified by respecting certain constraints. The analysis is carried out in a very small finite dimensional space, but it can be extended to more accurate finite-dimensional representations of the 2D Laplacian, with an increase of computational complexity. The aim of this paper is to introduce the preliminary steps in view of more serious generalizations.

Global Existence Structure of Parameters for Positive Solutions of a Singular $$(p_1,p_2)$$ ( p 1 , p 2 ) -Laplacian System

We study the homogeneous Dirichlet boundary value problem of a singular \((p_1,p_2)\)-Laplacian system with multiparameters. The global structure of multiparameters for existence, nonexistence and multiplicity of positive solutions can be obtained. Proofs are mainly based on the upper and lower solution method, the fixed point index theory in a cone, generalized Picone identity, global continuum theorem and degree arguments. As an application, we can establish the global result of the positive radial and decaying solutions for a class of quasilinear elliptic systems in exterior domains.