Semilinear Parabolic Equations Research Articles

Multilevel Picard approximations of high-dimensional semilinear partial differential equations with locally monotone coefficient functions

The full history recursive multilevel Picard approximation method for semilinear parabolic partial differential equations (PDEs) is the only method which provably overcomes the curse of dimensionality for general time horizons if the coefficient functions and the nonlinearity are globally Lipschitz continuous and the nonlinearity is gradient-independent. In this article we extend this result to locally monotone coefficient functions. Our results cover a range of semilinear PDEs with polynomial coefficient functions.

On a semilinear parabolic problem with non-local (Bitsadze–Samarskii type) boundary conditions in more dimensions

This paper studies a semilinear parabolic equation in a bounded domain Ω⊂Rd along with nonlocal boundary conditions. The boundary values are linked to the values of a solution on an interior (d−1)-dimensional manifold lying inside Ω. Firstly, the solvability of a steady-state problem is addressed. Secondly, involving the semi-discretization in time, a constructive algorithm for approximation of a solution to a transient problem is developed. The well-posedness of the problem in a weighted Hilbert space is shown. Convergence of approximations is addressed and the error estimated are derived. Numerical experiments support the theoretical algorithms.

A New Lagrange Multiplier Approach for Constructing Structure Preserving Schemes, II. Bound Preserving

In the second part of this series, we use the Lagrange multiplier approach proposed in the first part [Comput. Methods Appl. Mech. Engr., 391 (2022), 114585] to construct efficient and accurate bound and/or mass preserving schemes for a class of semilinear and quasi-linear parabolic equations. We establish stability results under a general setting and carry out an error analysis for a second-order bound preserving scheme with a hybrid spectral discretization in space. We apply our approach to several typical PDEs which preserve bound and/or mass and also present ample numerical results to validate our approach.

A necessary and sufficient condition for the existence of global solutions to discrete semilinear parabolic equations on networks

A necessary-sufficient condition for the existence and non-existence of global solutions to the following semilinear parabolic equationsut=Δu+ψtup,inΩ×0t∗,has remained as an open problem for a few decades. This paper discuss the necessary-sufficient condition for the existence and non-existence of global solutions of the above equations on networks for more general source term ψ(t)f(u) under various boundary conditions. The purpose of this paper is to prove that there is no global solution for any initial data if and only if the function f satisfies∫0∞ψ(t)fε∥S(t)u0∥∞∥S(t)u0∥∞dt=∞for every ε>0, where (S(t))t≥0 is the heat semigroup on networks with the corresponding boundary condition.

Partial asymptotic null controllability for semilinear evolution equations in Hilbert spaces

This work deals with the problem of partial asymptotic null controllability for constrained systems governed by semi-linear evolution equations, where the linear part generates a strongly continuous compact semigroup on Hilbert spaces. This consists of driven only a desired part of the system's state to the origin, taking into account mixed state-input constraints. We give sufficient conditions for the existence of appropriate control laws to ensure partial asymptotic null controllability. The explicit expression of such controls is given as strongly-weakly continuous selections of a set-valued map which is defined through a certain practical tangential condition. Application to semilinear parabolic partial differential equations is treated in order to illustrate the results established.

Continuous Differentiability of the Value Function of Semilinear Parabolic Infinite Time Horizon Optimal Control Problems on L2(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\Omega )$$\\end{document} Under Control Constraints

An abstract framework guaranteeing the local continuous differentiability of the value function associated with optimal stabilization problems subject to abstract semilinear parabolic equations subject to a norm constraint on the controls is established. It guarantees that the value function satisfies the associated Hamilton–Jacobi–Bellman equation in the classical sense. The applicability of the developed framework is demonstrated for specific semilinear parabolic equations.

Up to fourth-order unconditionally structure-preserving parametric single-step methods for semilinear parabolic equations

We propose and analyze a class of temporal up to fourth-order unconditionally structure-preserving single-step methods for Allen–Cahn-type semilinear parabolic equations. We first revisit some up to second-order exponential time different Runge–Kutta (ETDRK) schemes, and provide unified proofs for the unconditionally maximum-principle-preserving and mass-conserving properties. Noting that the stabilized ETDRK schemes belong to a special class of parametric Runge–Kutta schemes, we introduce the stabilized integrating factor Runge–Kutta (sIFRK) formulation to construct new high-order parametric single-step methods, and propose two strategies to eliminate the exponential effects of sIFRK: (1) a recursive approximation; (2) a combination of exponential and linear functions. Together with the nonnegativity of coefficients and non-decreasing of abscissas, the resulting two families of improved stabilized integrating factor Runge–Kutta (isIFRK) schemes can unconditionally preserve the maximum-principle and conserve the mass. The order conditions, linear stability and convergence in the l∞-norm are analyzed rigorously. We demonstrate that the proposed framework, which is explicit and free of limiters or cut-off post-processing, offers a simple, practical, and effective approach to developing high-order unconditionally structure-preserving algorithms. Comparisons with traditional schemes demonstrate the necessity of developing high-order unconditionally structure-preserving schemes. A series of numerical experiments verify theoretical results of proposed isIFRK schemes.

Bulk–surface Lie splitting for parabolic problems with dynamic boundary conditions

AbstractThis paper studies bulk–surface splitting methods of first order for (semilinear) parabolic partial differential equations with dynamic boundary conditions. The proposed Lie splitting scheme is based on a reformulation of the problem as a coupled partial differential–algebraic equation system, i.e., the boundary conditions are considered as a second dynamic equation that is coupled to the bulk problem. The splitting approach is combined with bulk–surface finite elements and an implicit Euler discretization of the two subsystems. We prove first-order convergence of the resulting fully discrete scheme in the presence of a weak CFL condition of the form $\tau \leqslant c h$ for some constant $c>0$. The convergence is also illustrated numerically using dynamic boundary conditions of Allen–Cahn type.

Composition operators on Herz-type Triebel–Lizorkin spaces with application to semilinear parabolic equations

Let $$G:\mathbb {R}\rightarrow \mathbb {R}$$ be a continuous function. In the first part of this paper, we investigate sufficient conditions on G such that $$\begin{aligned} \{G(f):f\in {\dot{K}}_{p,q}^{\alpha }F_{\beta }^{s}\}\subset {\dot{K}} _{p,q}^{\alpha }F_{\beta }^{s} \end{aligned}$$ holds. Here $${\dot{K}}_{p,q}^{\alpha }F_{\beta }^{s}$$ are Herz-type Triebel–Lizorkin spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights, Sobolev and Triebel–Lizorkin spaces of power weights. In the second part of this paper we will study local and global Cauchy problems for the semilinear parabolic equations $$\begin{aligned} \partial _{t}u-\Delta u=G(u) \end{aligned}$$ with initial data in Herz-type Triebel–Lizorkin spaces. Our results cover the results obtained with initial data in some known function spaces such us fractional Sobolev spaces. Some limit cases are given.

Semilinear parabolic equations in Herz spaces

In this paper, we will study local and global Cauchy problems for the semilinear parabolic equations with initial data in Herz spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights. Our results cover the results obtained with initial data in Lebesgue spaces. Moreover, the results in Herz spaces are a little different from the results in Lebesgue spaces.

Bounds for blow-up time in a semilinear parabolicproblem with variable exponents

This report deals with a blow-up of the solutions to a class of semilinear parabolic equations with variable exponents nonlinearities. Under some appropriate assumptions on the given data, a more general lower bound for a blow-up time is obtained if the solutions blow up. This result extends the recent results given by Baghaei Khadijeh et al. \cite{Baghaei}, which ensures the lower bounds for the blow-up time of solutions with initial data $\varphi\left( 0\right) =\int_{\Omega }u_{0}{}^{k}dx$, $k$ = constant.

Global solutions for semilinear rough partial differential equations

We construct global-in-time solutions for semilinear parabolic rough partial differential equations. We work on a scale of Banach spaces tailored to the controlled rough path approach and derive suitable a priori estimates of the solution which do not contain quadratic terms.

Unique continuation inequalities for the parabolic-elliptic chemotaxis system

This paper studies the quantitative unique continuation for a semi-linear parabolic-elliptic coupled system on a bounded domain Ω. This system is a simplified version of the chemotaxis model introduced by Keller and Segel in [14]. With the aid of priori L∞-estimates (for solutions of the system) built up in this paper, we treat the semi-linear parabolic equation in the system as a linear parabolic equation, and then use the frequency function method and the localization technique to build up two unique continuation inequalities for the system. As a consequence of the above-mentioned two inequalities, we have the following qualitative unique continuation property: if one component of a solution vanishes in a nonempty open subset ω⊂Ω at some time T>0, then the solution is identically zero.

Validated forward integration scheme for parabolic PDEs via Chebyshev series

In this paper we introduce a new approach to compute rigorously solutions of Cauchy problems for a class of semi-linear parabolic partial differential equations. Expanding solutions with Chebyshev series in time and Fourier series in space, we introduce a zero finding problem F(a)=0 on a Banach algebra X of Fourier–Chebyshev sequences, whose solution solves the Cauchy problem. The challenge lies in the fact that the linear part L=defDF(0) has an infinite block diagonal structure with blocks becoming less and less diagonal dominant at infinity. We introduce analytic estimates to show that L is an invertible linear operator on X, and we obtain explicit, rigorous and computable bounds for the operator norm ‖L−1‖B(X). These bounds are then used to verify the hypotheses of a Newton–Kantorovich type argument which shows that the (Newton-like) operator T(a)=defa−L−1F(a) is a contraction on a small ball centered at a numerical approximation of the Cauchy problem. The contraction mapping theorem yields a fixed point which corresponds to a classical (strong) solution of the Cauchy problem. The approach is simple to implement, numerically stable and is applicable to a class of PDE models, which include for instance Fisher’s equation and the Swift–Hohenberg equation. We apply our approach to each of these models.

Optimal Control of Semilinear Parabolic Equations with Non-smooth Pointwise-Integral Control Constraints in Time-Space

This work concentrates on a class of optimal control problems for semilinear parabolic equations subject to control constraint of the form Vert u(t)Vert _{L^1(varOmega )} le gamma for t in (0,T). This limits the total control that can be applied to the system at any instant of time. The L^1-norm of the constraint leads to sparsity of the control in space, for the time instants when the constraint is active. Due to the non-smoothness of the constraint, the analysis of the control problem requires new techniques. Existence of a solution, first and second order optimality conditions, and regularity of the optimal control are proved. Further, stability of the optimal controls with respect to gamma is investigated on the basis of different second order conditions.

Galerkin finite element method for a semi-linear parabolic equation with integral conditions

The present paper is devoted to prove the existence and uniquennes of a weak solution of a semi-linear reaction-difusion equation with only integral terms in the boundaries by using the finite element method and a prioryestimate.

New results concerning the hierarchical control of linear and semilinear parabolic equations

This paper deals with the application of multiple strategies to control some parabolic PDEs. We assume that we can act on the system through a hierarchy of distributed controls: with a first control (a follower), we drive the state exactly to zero; then, with an additional control (the leader), we minimize a prescribed cost functional. That means that we invert the roles played by leaders and followers in the recent literature. We study linear and semilinear problems. More precisely, we prove the existence (and uniqueness in the linear case) of a leader-follower couple. Then, we deduce an appropriate optimality system that must be satisfied by the controls and the corresponding state and adjoint states. We also indicate some generalizations to other controls, PDEs and systems. In particular, we establish similar existence and optimality results for hierarchical-biobjective (Pareto-Stackelberg) control problems, where there are two cost functionals and two independent leader controls whose main task is to find an associated Pareto equilibrium and one common follower in charge of null controllability.

Superconvergence for optimal control problems governed by semilinear parabolic equations

<abstract><p>In this paper, we first investigate optimal control problem for semilinear parabolic and introduce the standard $ L^2(\Omega) $-orthogonal projection and the elliptic projection. Then we present some necessary intermediate variables and their error estimates. At last, we derive the error estimates between the finite element solutions and $ L^2 $-orthogonal projection or the elliptic projection of the exact solutions.</p></abstract>

Leaderless Consensus Control of Nonlinear PIDE-Type Multi-Agent Systems With Time Delays

This paper studies leaderless consensus of semi-linear parabolic partial integro-differential equations based multi-agent systems (PIDEMASs) with time delays. Making use of the information interaction and coordination among the neighboring agents, consensus control of the leaderless PIDEMAS is constructed. Consensus of the leaderless PIDEMAS is analyzed by using a Lyapunov approach. Dealing with time-invariant delays and time-varying delays, two sufficient conditions for consensus of the leaderless PIDEMAS are respectively obtained in terms of LMIs. Two examples illustrate the effectiveness of developed theoretical results.

Global solutions of semilinear parabolic equations with drift term on Riemannian manifolds

<p style='text-indent:20px;'>We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term <inline-formula><tex-math id="M1">\begin{document}$ u^p $\end{document}</tex-math></inline-formula>, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that, for any <inline-formula><tex-math id="M2">\begin{document}$ p>1 $\end{document}</tex-math></inline-formula>, global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained for any <inline-formula><tex-math id="M3">\begin{document}$ p>1 $\end{document}</tex-math></inline-formula>, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast. We show that for any non trivial initial datum, for certain <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> depending on the Ricci curvature bound, global solutions cannot exist. On the other hand, for certain values of <inline-formula><tex-math id="M5">\begin{document}$ p $\end{document}</tex-math></inline-formula>, depending on the vector field <inline-formula><tex-math id="M6">\begin{document}$ b $\end{document}</tex-math></inline-formula>, global solutions exist, for sufficiently small initial data.</p>