Homogeneous Dirichlet Research Articles

Blow-Up Behaviors of Solutions to Reaction-Diffusion Equations With Nonlocal Sources and Variable Exponents

The blow-up problems of the solution are considered for a reaction-diffusion equation with nonlocal source and variable exponent. Firstly, the local existence and uniqueness of solutions to problem is proved under the help of the fixed point theorem. Secondly, by using the super- and sub-solution method, we determine some sufficient conditions for the occurrence of finite time blow-up under the homogeneous Dirichlet boundary conditions, i.e. the variable exponent is positive and the initial data is large enough. Moreover, the estimates of upper and lower bounds of blow-up time are given.

Cauchy problem for non-autonomous fractional evolution equations with nonlocal conditions of order $ (1, 2) $

<abstract><p>This article contracts through Cauchy problems in infinite-dimensional Banach spaces towards a system of nonlinear non-autonomous mixed type integro-differential fractional evolution equation by nonlocal conditions through noncompactness measure (MNC). We demonstrate the existence of novel mild solutions in the condition that the nonlinear function mollifies generally adequate, an MNC form and local growth form, using evolution families and fractional calculus theory, as well as the fixed-point theorem w.r.t. K-set-contractive operator and another MNC assessment procedure. Our findings simplify and improve upon past findings in this area. Finally, towards the end of this article, as an example of submissions, we use a fractional non-autonomous partial differential equation (PDE) with nonlocal conditions and a homogeneous Dirichlet boundary condition.</p></abstract>

The principal transmission condition

<abstract><p>The paper treats pseudodifferential operators $ P = \operatorname{Op}(p(\xi)) $ with homogeneous complex symbol $ p(\xi) $ of order $ 2a > 0 $, generalizing the fractional Laplacian $ (-\Delta)^a $ but lacking its symmetries, and taken to act on the halfspace ${\mathbb R}^n_+$. The operators are seen to satisfy a principal $ \mu $-transmission condition relative to ${\mathbb R}^n_+$, but generally not the full $ \mu $-transmission condition satisfied by $ (-\Delta)^a $ and related operators (with $ \mu = a $). However, $ P $ acts well on the so-called $ \mu $-transmission spaces over ${\mathbb R}^n_+$ (defined in earlier works), and when $ P $ moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for $ P $, leading to regularity results with a factor $ x_n^\mu $ (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over ${\mathbb R}^n_+$ for $ P $ acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond certain operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, as new results for these operators. Since the principal $ \mu $-transmission condition has weaker requirements than the full $ \mu $-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.</p></abstract>

Global well-posedness and exponential decay for 3D nonhomogeneous magneto-micropolar fluid equations with vacuum

<p style='text-indent:20px;'>We consider an initial boundary value problem of three-dimensional (3D) nonhomogeneous magneto-micropolar fluid equations in a bounded simply connected smooth domain with homogeneous Dirichlet boundary conditions for the velocity and micro-rotational velocity and Navier-slip boundary condition for the magnetic field. We prove the global existence and exponential decay of strong solutions provided that some smallness condition holds true. Note that although the system degenerates near vacuum, there is no need to require compatibility conditions for the initial data via time weighted techniques.</p>

Reaction-advection-diffusion competition models under lethal boundary conditions

<p style='text-indent:20px;'>In this study, we consider a Lotka–Volterra reaction–diffusion–advection model for two competing species under homogeneous Dirichlet boundary conditions, describing a hostile environment at the boundary. In particular, we deal with the case in which one species diffuses at a constant rate, whereas the other species has a constant rate diffusion rate with a directed movement toward a better habitat in a heterogeneous environment with a lethal boundary. By analyzing linearized eigenvalue problems from the system, we conclude that the species dispersion in the advection direction is not always beneficial, and survival may be determined by the convexity of the environment. Further, we obtain the coexistence of steady-states to the system under the instability conditions of two semi-trivial solutions and the uniqueness of the coexistence steady states, implying the global asymptotic stability of the positive steady-state.</p>

Bounded nonnegative weak solutions to anisotropic parabolic double phase problems with variable growth

We study the homogeneous Dirichlet problem for a class of nonlinear anisotropic parabolic double phase equations with nonstandard growth conditions. We prove the existence of a unique nonnegative weak solution in suitable Orlicz-Sobolev spaces, and derive the global boundedness of the weak solution.

Multiplicity of solutions to class of nonlocal elliptic problems with critical exponents

In this paper, we establish existence of infinitely many weak solutions for a class of quasilinear stationary Kirchhoff‐type equations, which involves a general variable exponent elliptic operator with critical growth. Precisely, we study the following nonlocal problem: where Ω is a bounded smooth domain of , with homogeneous Dirichlet boundary conditions on ∂Ω, the nonlinearity is a continuous function, is a function of the class is a continuous function, whose properties will be introduced later, λ is a positive parameter and . We assume that , where is the critical Sobolev exponent. We will prove that the problem has infinitely many solutions and also we obtain the asymptotic behavior of the solution as λ → 0+. Furthermore, we emphasize that a difference with previous researches is that the conditions on a(·) are general overall enough to incorporate some interesting differential operators. Our work covers a feature of the Kirchhoff's problems, that is, the fact that the Kirchhoff's function M in zero is different from zero, it also covers a wide class of nonlocal problems for p(x) > 1, for all . The main tool to find critical points of the Euler–Lagrange functional associated with this problem is through a suitable truncation argument, concentration‐compactness principle for variable exponent found in Bonder and Silva (2010), and the genus theory introduced by Krasnoselskii. The result of this paper extends or complements, or else completes recent papers and is new in several directions for the stationary Kirchhoff equations involving the p(x)‐Laplacian type operators.

Boundary regularity for elliptic systems with p,q-growth

We investigate the boundary regularity of minimizers of convex integral functionals with nonstandard p,q-growth and with Uhlenbeck structure. We consider arbitrary convex domains Ω and homogeneous Dirichlet data on some part Γ⊂∂Ω of the boundary. For the integrand we assume only a non-standard p,q-growth condition. We establish Lipschitz regularity of minimizers up to Γ, provided the gap between the growth exponents p and q is not too large, more precisely if 1<p≤q<p(1+2n). To our knowledge, this is the first boundary regularity result under a non-standard p,q-growth condition.

Optimal identification of cavities in the Generalized Plane Stress problem in linear elasticity

For the Generalized Plane Stress (GPS) problem in linear elasticity, we obtain an optimal stability estimate of logarithmic type for the inverse problem of determining smooth cavities inside a thin isotropic cylinder from a single boundary measurement of traction and displacement. The result is obtained by reformulating the GPS problem as a Kirchhoff–Love plate-like problem in terms of the Airy function, and by using the strong unique continuation at the boundary for a Kirchhoff–Love plate operator under homogeneous Dirichlet conditions, which has recently been obtained in [G. Alessandrini et al., Arch. Ration. Mech. Anal. 231 (2019)].

Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks

In this paper, we introduce a new approach based on distance fields to exactly impose boundary conditions in physics-informed deep neural networks. The challenges in satisfying Dirichlet boundary conditions in meshfree and particle methods are well-known. This issue is also pertinent in the development of physics informed neural networks (PINN) for the solution of partial differential equations. We introduce geometry-aware trial functions in artificial neural networks to improve the training in deep learning for partial differential equations. To this end, we use concepts from constructive solid geometry (R-functions) and generalized barycentric coordinates (mean value potential fields) to construct ϕ(x), an approximate distance function to the boundary of a domain in Rd. To exactly impose homogeneous Dirichlet boundary conditions, the trial function is taken as ϕ(x) multiplied by the PINN approximation, and its generalization via transfinite interpolation is used to a priori satisfy inhomogeneous Dirichlet (essential), Neumann (natural), and Robin boundary conditions on complex geometries. In doing so, we eliminate modeling error associated with the satisfaction of boundary conditions in a collocation method and ensure that kinematic admissibility is met pointwise in a Ritz method. With this new ansatz, the training for the neural network is simplified: sole contribution to the loss function is from the residual error at interior collocation points where the governing equation is required to be satisfied. Numerical solutions are computed using strong form collocation and Ritz minimization. To convey the main ideas and to assess the accuracy of the approach, we present numerical solutions for linear and nonlinear boundary-value problems over convex and nonconvex polygonal domains as well as over domains with curved boundaries. Benchmark problems in one dimension for linear elasticity, advection-diffusion, and beam bending; and in two dimensions for the steady-state heat equation, Laplace equation, biharmonic equation (Kirchhoff plate bending), and the nonlinear Eikonal equation are considered. The construction of approximate distance functions using R-functions extends to higher dimensions, and we showcase its use by solving a Poisson problem with homogeneous Dirichlet boundary conditions over the four-dimensional hypercube. The proposed approach consistently outperforms a standard PINN-based collocation method, which underscores the importance of exactly (a priori) satisfying the boundary condition when constructing a loss function in PINN. This study provides a pathway for meshfree analysis to be conducted on the exact geometry without domain discretization.

On fractional p-Laplacian type equations with general nonlinearities

In this paper, we study the existence and multiplicity of solutions for a class of quasi-linear elliptic problems driven by a nonlocal integro-differential operator with homogeneous Dirichlet boundary conditions. As a particular case, we study the following problem: \begin{equation*} \left\{ \begin{array}{l} (-\Delta)_p^s u= f(x,u) \quad \hfill \textrm{in} \ \Omega,\\ \quad u=0 \ \hfill \textrm{in} \ R^N \setminus \Omega, \end{array} \right.\\ \end{equation*} where $(-\Delta)_p^s$ is the fractional p-Laplacian operator, $\Omega$ is an open bounded subset of $R^N$ with Lipschitz boundary and $f:\Omega \times R \to R$ is a generic Carath\'eodory function satisfying either a $p-$sublinear or a $p-$superlinear growth condition.

Spectrum of Fractional and Fractional Prabhakar Sturm–Liouville Problems with Homogeneous Dirichlet Boundary Conditions

In this study, we consider regular eigenvalue problems formulated by using the left and right standard fractional derivatives and extend the notion of a fractional Sturm–Liouville problem to the regular Prabhakar eigenvalue problem, which includes the left and right Prabhakar derivatives. In both cases, we study the spectral properties of Sturm–Liouville operators on function space restricted by homogeneous Dirichlet boundary conditions. Fractional and fractional Prabhakar Sturm–Liouville problems are converted into the equivalent integral ones. Afterwards, the integral Sturm–Liouville operators are rewritten as Hilbert–Schmidt operators determined by kernels, which are continuous under the corresponding assumptions. In particular, the range of fractional order is here restricted to interval (1/2,1]. Applying the spectral Hilbert–Schmidt theorem, we prove that the spectrum of integral Sturm–Liouville operators is discrete and the system of eigenfunctions forms a basis in the corresponding Hilbert space. Then, equivalence results for integral and differential versions of respective eigenvalue problems lead to the main theorems on the discrete spectrum of differential fractional and fractional Prabhakar Sturm–Liouville operators.

Numerical approximations of coupled forward–backward SPDEs

We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward–backward stochastic partial differential equations (FBSPDEs) with homogeneous Dirichlet boundary conditions. Precisely, we generalize the pioneering work of Dunst and Prohl [SIAM J. Sci. Comp., 38(2017), 2725–2755] by considering general nonlinear and nonlocal FBSPDEs with more inclusive coupling; self-contained proofs are provided and different numerical techniques for the resulting finite dimensional equations are adopted. For such FBSPDEs allowing for non-Lipschitz nonlinearity on unknown variables, we prove the existence and uniqueness of the strong solution as well as of the weak solution. Then, the finite element method in the spatial domain leads to approximations of FBSPDEs by finite-dimensional forward–backward stochastic differential equations which are numerically computed by using some deep learning-based schemes. A strong convergence analysis is addressed for the spatial discretization of coupled FBSPDEs under an additional Lipschitz assumption, and the numerical examples, including both decoupled and coupled cases, indicate that our methods are quite efficient.

The Meyers Estimates for Domains Perforated along the Boundary

In this paper, we consider an elliptic problem in a domain perforated along the boundary. By setting a homogeneous Dirichlet condition on the boundary of the cavities and a homogeneous Neumann condition on the outer boundary of the domain, we prove higher integrability of the gradient of the solution to the problem.

Global Existence of Solution for the Fisher Equation via Faedo–Galerkin’s Method

In this study, we consider the Fisher equation in bounded domains. By Faedo–Galerkin’s method and with a homogeneous Dirichlet conditions, the existence of a global solution is proved.

An enhancement of the fast time-domain boundary element method for the three-dimensional wave equation

Our objective is to stabilise and accelerate the time-domain boundary element method (TDBEM) for the three-dimensional wave equation. To overcome the potential time instability, we considered using the Burton–Miller-type boundary integral equation (BMBIE) instead of the ordinary boundary integral equation (OBIE), which consists of the single- and double-layer potentials. In addition, we introduced a smooth temporal basis, i.e. the B-spline temporal basis of order d, whereas d=1 was used together with the OBIE in a previous study [1]. Corresponding to these new techniques, we generalised the interpolation-based fast multipole method that was developed in [1]. In particular, we constructed the multipole-to-local formula (M2L) so that even for d≥2 we can maintain the computational complexity of the entire algorithm, i.e. O(Ns1+δNt), where Ns and Nt denote the number of boundary elements and the number of time steps, respectively, and δ is theoretically estimated as 1/3 or 1/2. The numerical examples indicated that the BMBIE is indispensable for solving the homogeneous Dirichlet problem, but the order d cannot exceed 1 owing to the doubtful cancellation of significant digits when calculating the corresponding layer potentials. In regard to the homogeneous Neumann problem, the previous TDBEM based on the OBIE with d=1 can be unstable, whereas it was found that the BMBIE with d=2 can be stable and accurate. The present study will enhance the usefulness of the TDBEM for 3D scalar wave problems.

On the seven-diagonals splitting for the cubic spline wavelets with six vanishing moments on an interval

This study uses a zeroing property of the first six moments for constructing a splitting algorithm for the cubic spline wavelets. First, we construct a system of cubic basic spline-wavelets, realizing orthogonal conditions to all polynomials up to any degree. Then, using the homogeneous Dirichlet boundary conditions, we adapt spaces to the orthogonality to all polynomials up to the fifth degree on the closed interval. The originality of the study consists of obtaining implicit finite relations connecting the coefficients of the spline decomposition at the initial scale with the spline coefficients and wavelet coefficients at the nested scale by a tape system of linear algebraic equations with a non-degenerate matrix. After excluding the even rows of the system, the resulting transformation matrix has seven diagonals, instead of five as in the previous case with four zero moments. A modification of the system is performed, which ensures a strict diagonal dominance, and, consequently, the stability of the calculations. The comparative results of numerical experiments on approximating and calculating the derivatives of a discrete function are presented.

Global dynamic bifurcation of local semiflows and nonlinear evolution equations

In this work new global dynamic bifurcation theorems are established for local semiflows on complete metric spaces. These theorems are applied to the nonlinear evolution equation u t + A u = f λ ( u ) in a Banach space X , where A is a sectorial operator with compact resolvent, and f λ ( 0 ) = 0 for all λ ∈ R . In particular, if f λ takes the form f λ ( u ) = λ u + f ( u ) , it is shown that the global dynamic bifurcation branch of a bifurcation point ( 0 , λ 0 ) is necessarily unbounded without assuming the “crossing odd-multiplicity” condition. It has long been recognized that dynamical methods can be helpful in solving static problems. As an example, the bifurcation of elliptic equations on bounded domains associated with homogenous Dirichlet boundary condition is addressed. By considering the corresponding nonclassical parabolic flows and applying the global dynamic bifurcation theorems given here for local semiflows, some new results with global features are derived.

Blow-Up Results for a Class of Quasilinear Parabolic Equation with Power Nonlinearity and Nonlocal Source

This paper deals with a class of quasilinear parabolic equation with power nonlinearity and nonlocal source under homogeneous Dirichlet boundary condition in a smooth bounded domain; we obtain the blow-up condition and blow-up results under the condition of nonpositive initial energy.

The location of hot spots and other extremal points

In a domain of the Euclidean space, we estimate from below the distance to the boundary of global maximum points of solutions of elliptic and parabolic equations with homogeneous Dirichlet boundary values. As reference cases, we consider the torsional rigidity function of a bar, the first mode of a vibrating membrane, and the temperature of a heat conductor grounded to zero at the boundary. Our main results are presented for domains with mean convex boundary and compare that distance to the inradius of the relevant domain. For the torsional rigidity function, the obtained bound only depends on the space dimension. The more general case of a boundary which is not mean convex is also considered. However, in this case the estimates also depend on some geometrical quantities such as the diameter and the radius of the largest exterior osculating ball to the relevant domain, or the minimum of the mean curvature of the boundary. For the first mode, the relevant bound only depends on the space dimension, as well. Moreover, it largely improves on an earlier estimate obtained for convex domains by the first author and co-authors. The bound related to the temperature depends on time and the initial distribution of temperature. Such a bound is substantially consistent with what one obtains in the stationary situation. The methods employed are based on elementary arguments and existing literature, and can be extended to other situations that entail quasilinear equations, isotropic and anisotropic, and also certain classes of semilinear equations.