Nonlocal Heat Equation Research Articles

On positive solutions of the Cauchy problem for doubly nonlocal equations

We study the Cauchy problem for the doubly nonlocal equation where and for some m ∈ ℝ and 0 < α ≤ +∞, here α = +∞ means that J is compactly supported. The problem not only describes nonlocal diffusions and nonlocal interactions, but also it is a nonlocal counterpart of the nonlocal heat equation in [10] where and m > 0. As for the local existence of positive solutions, we find that problem (0.1) is quite different from the nonlocal heat equation. That is, problem (0.1) admits positive solutions if and only if m > n − q(n + α), however the nonlocal heat equation has positive solutions for any m ∈ ℝ. For m ≥ 0 or n(1 − q) < m < 0 with α = +∞, we determine the critical Fujita curve of problem (0.1) as F(p, q, n, m, r, α) := (p − 1)[nq − (n − m)+] − q(β − nr) = 0 with β = min{α, 2}. That is, if F ≤ 0, every positive solution of (0.1) blows up in finite time, whereas if F > 0, there exist both nonglobal solutions and global solutions. In the supercritical case F > 0, we further discuss the second critical exponent which describes the critical decay rate of initial data to distinguish both solutions. Particularly, we find a new phenomenon, namely, if with α = +∞, the second critical exponent is independent of p and r.

Blow-up solution to an abstract non-local stochastic heat equation with Lévy noise

This paper investigates explosive solutions to a class of non-local stochastic heat equations driven by Lévy noise. We establish sufficient conditions to show that the positive solution will blow-up in finite time.

Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection

We study the large-time behaviour of nonnegative solutions to the Cauchy problem for a nonlocal heat equation with a nonlinear convection term. The diffusion operator is the infinitesimal generator of a stable Lévy process, which may be highly anisotropic. The initial data are assumed to be bounded and integrable. The mass of the solution is conserved along the evolution, and the large-time behaviour is given by the source-type solution, with the same mass, of a limit equation that depends on the relative strength of convection and diffusion. When diffusion is stronger than convection the original equation simplifies asymptotically to the purely diffusive nonlocal heat equation. When convection dominates, it does so only in the direction of convection, and the limit equation is still diffusive in the subspace orthogonal to this direction, with a diffusion operator that is a “projection” of the original one onto the subspace. The determination of this projection is one of the main issues of the paper. When convection and diffusion are of the same order the limit equation coincides with the original one.Most of our results are new even in the isotropic case in which the diffusion operator is the fractional Laplacian. We are able to cover both the cases of slow and fast convection, as long as the mass is preserved. Fast convection, which corresponds to convection nonlinearities that are not locally Lipschitz, but only locally Hölder, has not been considered before in the nonlocal diffusion setting.

Photothermal effects of semiconducting medium with non-local theory

A generalized magnetothermoelastic half-space problem is considered for a homogeneous, isotropic and semiconducting medium under the non-local heat equation with dual-phase-lag model. The boundary surface of the medium is subjected to a prescribed time-dependent and exponential order compression along with a prescribed temperature and carrier intensity gradient. Two integral transformations, Laplace transform for time variable and Fourier transform for space variable, are employed to the equations of motion and the heat conduction equation for formulation of a vector-matrix differential equation which is then solved by using eigenvalue approach. Inversion processes for the two integral transforms are carried out numerically. Finally, the effects of physical field variables and stress components are analyzed and illustrated graphically under the variation of different physical parameters.

Second-order accurate, robust and efficient ADI Galerkin technique for the three-dimensional nonlocal heat model arising in viscoelasticity

This paper adopts an accurate and robust algorithm for the nonlocal heat equation having a weakly singular kernel in three dimensions. The proposed method approximates the unknown solution in two steps. First, the temporal discretization is achieved through a Crank-Nicolson finite difference with the nonuniform temporal meshes, which are applied to tackle the singularity behavior for the exact solution when t=0. Second, the spatial discretization is derived by means of the Galerkin finite element. Additionally, the alternating direction implicit (ADI) approach is adopted to decrease computational burden. The proposed method in terms of the convergence and stability analysis is studied by using the discrete energy method in detail with optimal rates of convergence O(k2+hr+1), r≥1 based on some appropriate assumptions, where k and h represent step sizes in the temporal and spatial directions, respectively. Numerical results highlight the effectiveness of the proposed method and the correctness of the theoretical prediction.

Recovering Source Term and Temperature Distribution for Nonlocal Heat Equation

We consider two problems of recovering the source terms along with heat concentration for a time fractional heat equation involving the so-called mth level fractional derivative (LFD) (proposed in a paper by Luchko [1]) in time variable of order between 0 and 1. The solutions of both problems are obtained by using eigenfunction expansion method. The series solutions of the inverse problems are proved to be unique and regular. The ill-posedness of inverse problems is proved in the sense of Hadamard and some numerical examples are presented.

Direct and inverse problems for nonlocal heat equation with boundary conditions of periodic type

A mathematical model of the process of heat diffusion in a closed metal wire is considered. This wire is wrapped around a thin sheet of insulation material. We assume that the insulation is slightly permeable. Because of this, the temperature at the point of the wire on one side of the insulation influences the diffusion process in the wire on the other side of the insulation. Thus, the standard heat equation will change and an extra term with involution will be added. When modeling of this process there arises an initial-boundary value problem for a one-dimensional heat equation with involution and with a boundary condition of periodic type with respect to a spatial variable. We prove the well-posedness of the formulated problem in the class of strong generalized solutions. The use of the method of separation of variables leads to a spectral problem for an ordinary differential operator with involution at the highest derivative. All eigenfunctions of the problem are constructed. In the case when all eigenvalues of the problem are simple, the system of eigenfunctions does not form an unconditional basis. A criterion when this spectral problem can have an infinite number of multiple eigenvalues is proved. Corresponding root subspaces consist of one eigenfunction and one associated function. We prove that the system of root functions forms an unconditional basis and can be used for constructing a solution of the heat conduction problem by the method of separation of variables. We also consider an inverse problem. This is the problem on restoring (simultaneously with solving) of an unknown stationary source of external influence with respect to an additionally known final state. The existence of a unique solution of this inverse problem and its stability with respect to initial and final data are proved.

ON ROOT FUNCTIONS OF NONLOCAL DIFFERENTIAL SECOND-ORDER OPERATOR WITH BOUNDARY CONDITIONS OF PERIODIC TYPE

In this paper we consider one class of spectral problems for a nonlocal ordinary differential operator (with involution in the main part) with nonlocal boundary conditions of periodic type. Such problems arise when solving by the method of separation of variables for a nonlocal heat equation. We investigate spectral properties of the problem for the nonlocal ordinary differential equation Ly (x) ≡ −y 00 (x) + εy00 (−x) = λy (x), −1 &lt; x &lt; 1. Here λ is a spectral parameter, |ε| &lt; 1. Such equations are called nonlocal because they have a term y 00 (−x) with involutional argument deviation. Boundary conditions are nonlocal y 0 (−1) + ay0 (1) = 0, y (−1) − y (1) = 0. Earlier this problem has been investigated for the special case a = −1. We consider the case a 6= −1. A criterion for simplicity of eigenvalues of the problem is proved: the eigenvalues will be simple if and only if the number r = sqrt{p (1 − ε) / (1 + ε)} is irrational. We show that if the number r is irrational, then all the eigenvalues of the problem are simple, and the system of eigenfunctions of the problem is complete and minimal but does not form an unconditional basis in L2(−1, 1). For the case of rational numbers r, it is proved that a (chosen in a special way) system of eigen- and associated functions forms an unconditional basis in L2(−1, 1).

Nonlocal thermal diffusion in one-dimensional periodic lattice

Heat conduction in micro-structured rods can be simulated with unidimensional periodic lattices. In this paper, scale effects in conduction problems, and more generally in diffusion problems, are studied from a one-dimensional thermal lattice. The thermal lattice is governed by a mixed differential-difference equation, accounting for some spatial microstructure. Exact solutions of the linear time-dependent spatial difference equation are derived for a thermal lattice under initial uniform temperature field with some temperature perturbations at the boundary. This discrete heat equation is approximated by a continuous nonlocal heat equation built by continualization of the thermal lattice equations. It is shown that such a nonlocal heat equation may be equivalently obtained from a nonlocal Fourier's law. Exact solutions of the thermal lattice problem (discrete heat equation) are compared with the ones of the local and nonlocal heat equation. An error analysis confirms the accurate calibration of the length scale of the nonlocal diffusion law with respect to the lattice spacing. The nonlocal heat equation may efficiently capture scale effect phenomena in periodic thermal lattices.

On Wiener's violent oscillations, Popov's curves, and Hopf's supercritical bifurcation for a scalar heat equation

AbstractA parameter‐dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non‐self‐adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter‐dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that admits an interpretation as a thermostat device or in the context of an agent‐based price formation model for a market. The existence and the stability of periodic self‐oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation are proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. They follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only, it can be extended naturally to arbitrary Euclidean dimension and to manifolds.

Decay/growth rates for inhomogeneous heat equations with memory. The case of large dimensions

&lt;abstract&gt;&lt;p&gt;We study the decay/growth rates in all $ L^p $ norms of solutions to an inhomogeneous nonlocal heat equation in $ \mathbb{R}^N $ involving a Caputo $ \alpha $-time derivative and a power $ \beta $ of the Laplacian when the dimension is large, $ N &amp;gt; 4\beta $. Rates depend strongly on the space-time scale and on the time behavior of the spatial $ L^1 $ norm of the forcing term.&lt;/p&gt;&lt;/abstract&gt;

Large-Time Behavior for a Fully Nonlocal Heat Equation

We study the large-time behavior in all Lp norms and in different space-time scales of solutions to a nonlocal heat equation in $\mathbb {R}^{N}$ involving a Caputo α-time derivative and a power of the Laplacian (−Δ)s, s ∈ (0,1), extending recent results by the authors for the case s = 1. The initial data are assumed to be integrable, and, when required, to be also in Lp. The main novelty with respect to the case s = 1 comes from the behaviour in fast scales, for which, thanks to the fat tails of the fundamental solution of the equation, we are able to give results that are not available neither for the case s = 1 nor, to our knowledge, for the standard heat equation, s = 1, α = 1.

Strong solutions to a nonlocal-in-time semilinear heat equation

The existence of strong solutions to a nonlocal semilinear heat equation is shown. The main feature of the equation is that the nonlocal term depends on the unknown on the whole time interval of existence, the latter being given a priori. The proof relies on Schauder’s fixed point theorem and semigroup theory.

Local Null-Controllability of a Nonlocal Semilinear Heat Equation

This paper deals with the problem of internal null-controllability of a heat equation posed on a bounded domain with Dirichlet boundary conditions and perturbed by a semilinear nonlocal term. We prove the small-time local null-controllability of the equation. The proof relies on two main arguments. First, we establish the small-time local null-controllability of a $$2 \times 2$$ reaction-diffusion system, where the second equation is governed by the parabolic operator $$\tau \partial _t - \sigma \varDelta $$, $$\tau , \sigma > 0$$. More precisely, this controllability result is obtained uniformly with respect to the parameters $$(\tau , \sigma ) \in (0,1) \times (1, + \infty )$$. Secondly, we observe that the semilinear nonlocal heat equation is actually the asymptotic derivation of the reaction-diffusion system in the limit $$(\tau ,\sigma ) \rightarrow (0,+\infty )$$. Finally, we illustrate these results by numerical simulations.

An asymptotically compatible formulation for local-to-nonlocal coupling problems without overlapping regions

In this paper we design and analyze an explicit partitioned procedure for a 2D dynamic local-to-nonlocal (LtN) coupling problem, based on a new nonlocal Robin-type transmission condition. The nonlocal subproblem is modeled by the nonlocal heat equation with a finite horizon parameter δ characterizing the range of nonlocal interactions, and the local subproblem is described by the classical heat equation. We consider a heterogeneous system where the local and nonlocal subproblems present different physical properties, and employ no overlapping region between the two subdomains. We first propose a new generalization of classical local Neumann-type condition by converting the local flux to a correction term in the nonlocal model, and show that the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ norm. We then extend the nonlocal Neumann-type boundary condition to a Robin-type boundary condition, and develop a local-to-nonlocal coupling formulation with Robin–Dirichlet transmission conditions. To stabilize the explicit coupling procedure and to achieve asymptotic compatibility, the choice of the coefficient in the Robin condition is obtained via amplification factor analysis for the discretized system with coarse grids. Employing a high-order meshfree discretization method in the nonlocal solver and a linear finite element method in the local solver, the selection of optimal Robin coefficients are verified with numerical experiments on heterogeneous and complicated domains. With the developed optimal coupling strategy, we numerically demonstrate the coupling framework’s asymptotic convergence to the local limit with an O(δ)=O(h) rate, when there is a fixed ratio between the horizon size δ and the spatial discretization size h.

Anisotropic nonlocal diffusion equations with singular forcing

We prove existence, uniqueness and regularity of solutions of nonlocal heat equations associated to anisotropic stable diffusion operators. The main features are that the right-hand side has very little regularity and that the spectral measure can be singular in some directions. The proofs require having good enough estimates for the corresponding heat kernels and their derivatives.

Decay estimate and non-extinction of solutions of p-Laplacian nonlocal heat equations

The main goal of this work is to study the initial boundary value problem of a nonlocal heat equations with logarithmic nonlinearity in a bounded domain. By using the logarithmic Sobolev inequality and potential wells method, we obtain the decay, blow-up and non-extinction of solutions under some conditions, and the results extend the results of a recent paper Lijun Yan and Zuodong Yang (2018).

Local properties for weak solutions of nonlocal heat equations

In this paper, using the De Giorgi–Nash–Moser theory, we obtain an interior parabolic Hölder regularity for weak solutions of nonlocal heat equations given by an integro-differential operator LK as follows; LKu+∂tu=0in Ω×(−T,0]u=gin ((Rn∖Ω)×(−T,0])∪(Ω×{t=−T})where g∈C(Rn×[−T,0])∩L∞(Rn×(−T,0])∩HTs(Rn) and Ω⊂Rn is a bounded domain with Lipschitz boundary. In addition, we get the local boundedness of such weak solutions.

Nonlocal Harnack inequalities for nonlocal heat equations

By applying the De Giorgi-Nash-Moser theory, we obtain nonlocal Harnack inequalities for locally nonnegative weak solutions of nonlocal parabolic equations given by an integro-differential operator LK as follows:{LKu+∂tu=0 in ΩI:=Ω×(−T,0] u=g in ∂pΩI:=((Rn∖Ω)×(−T,0])∪(Ω×{t=−T}) for g∈C(RI⁎n)∩L∞(Rn×(−T,0])∩HTs(Rn) and a bounded domain Ω⊂Rn with Lipschitz boundary. Interestingly, this result implies the classical Harnack inequalities for globally nonnegative weak solutions.

Blow-up solutions of the stochastic nonlocal heat equations

This paper is concerned with the blow-up phenomenon of stochastic nonlocal heat equations. We first establish the sufficient condition to ensure that the stochastic nonlocal heat equations have a unique non-negative solution. Then the problem of blow-up solutions in finite time is considered.