Semilinear Parabolic Equations Research Articles

Global existence and blow-up for semilinear parabolic equation with critical exponent in R N

In this paper, we use the self-similar transformation and the modified potential well method to study the long time behaviors of solutions to the classical semilinear parabolic equation associated with critical Sobolev exponent in R N . Global existence and finite time blowup of solutions are proved when the initial energy is in three cases. When the initial energy is low or critical, we not only give a threshold result for the global existence and blowup of solutions, but also obtain the decay rate of the L 2 norm for global solutions. When the initial energy is high, sufficient conditions for the global existence and blowup of solutions are also provided. We extend the recent results which were obtained in [R. Ikehata, M. Ishiwata, T. Suzuki, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(2010), No. 3, 877–900].

Differential Harnack estimate of solutions to a class of semilinear parabolic equation

Differential Harnack estimate of solutions to a class of semilinear parabolic equation

Finite-dimensional negatively invariant subsets of Banach spaces

We give a simple proof of a result due to Mañé (1981) [17] that a compact subset A of a Banach space that is negatively invariant for a map S is finite-dimensional if DS(x)=C(x)+L(x), where C is compact and L is a contraction (and both are linear). In particular, we show that if S is compact and differentiable then A is finite-dimensional. We also prove some results (following Málek et al. (1994) [15] and Zelik (2000) [23]) that give bounds on the (box-counting) dimension of such sets assuming a ‘smoothing property’: in its simplest form this requires S to be Lipschitz from X into another Banach space Z that is compactly embedded in X. The resulting bounds depend on the Kolmogorov ε-entropy of the embedding of Z into X. We give applications to an abstract semilinear parabolic equation and the two-dimensional Navier–Stokes equations on a periodic domain.

Superconvergence analysis of Galerkin method for semilinear parabolic integro-differential equation

In this paper, based on the bilinear element used for spatial discretization and a linearized backward Euler scheme used for temporal discretization, the superconvergence error estimate is derived for semilinear parabolic integro-differential equation without certain time-step restrictions. The key is to derive a uniform boundness of the numerical solution in energy norm under the weaker assumption compared to previous literatures for nonlinear term. Numerical results are presented to confirm the correctness of the theoretical analysis.

Generalization of the Trotter–Daletsky formula for systems of the "reaction–diffusion" type

An iterative method for constructing a solution to the Cauchy problem for a system of parabolic equations with a nonlinear potential has been proposed and substantiated. The method is based on the Trotter–Daletsky formula, generalized for a nonlinear perturbation of an elliptic operator. The idea of generalization is the construction of a composition of the semigroup generated by the Laplacian and the phase flow corresponding to a system of ordinary differential equations. A computational experiment performed for a two-dimensional system of semilinear parabolic equations of the “reaction–diffusion” type confirms estimates for the convergence of iterations established in the proof of this formula. Obtained results suggest the feasibility of an unconventional approach to modeling dynamic systems with distributed parameters.

Prescribed-time stabilization of semi-linear parabolic equations subject to distributed disturbance

In this paper, we investigate the problem of prescribed-time stabilization for a class of semilinear parabolic systems subject to spatiotemporal-varying disturbance via distributed control. By employing the time-varying feedback gain and disturbance suppression method, the proposed control law is continuous and stabilizes the closed-loop system within the prescribed time, where the convergence time is independent of initial values and can be given in advance as needed. When the upper bound of disturbance is known, we use a hyperbolic tangent function to restrain disturbance. While the upper bound of disturbance is unknown, we design the prescribed-time adaptive law and a prescribed-time disturbance observer estimating the disturbance itself. Some numerical examples are provided to verify the theoretical results.

Global existence and non-existence analyses for a semilinear edge degenerate parabolic equation with singular potential term

This paper discusses initial boundary value problem for a semilinear edge degenerate parabolic equation and corresponding stationary problem. We first find some initial conditions with different energy levels such that the solution exists globally and blows up in finite time, respectively. We also study the asymptotic behaviors like exponential decay and exponential growth for solution and energy function. Especially, we show the solution of evolution problem will converge to the steady state solution. Additionally, we find that there are two explicit vacuum regions which are ball and annulus respectively, that is to say, there is no solution belongs to them and all solutions are isolated by them. Finally, we discuss the existence of ground state solution to the stationary problem. The instability of the ground state solution is considered and we prove that there exists initial value such that the instability occurs as a blow-up in finite time.

On the existence and uniqueness of the solution of the coefficient inverse problem for a semilinear parabolic equation

In this paper, we consider the problem of determining the source function and the coefficient by the derivative with respect to time in a semilinear parabolic equation with overdetermination conditions defined on two different hyperplanes. The existence and uniqueness theorems of the classical solution of the posed coefficient inverse problem in the class of smooth bounded functions were proved. An example of input data satisfying the conditions of the proved theorems is given.

Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

We introduce a new family of numerical algorithms for approximating solutions of general high-dimensional semilinear parabolic partial differential equations at single space-time points. The algorithm is obtained through a delicate combination of the Feynman–Kac and the Bismut–Elworthy–Li formulas, and an approximate decomposition of the Picard fixed-point iteration with multilevel accuracy. The algorithm has been tested on a variety of semilinear partial differential equations that arise in physics and finance, with satisfactory results. Analytical tools needed for the analysis of such algorithms, including a semilinear Feynman–Kac formula, a new class of seminorms and their recursive inequalities, are also introduced. They allow us to prove for semilinear heat equations with gradient-independent nonlinearities that the computational complexity of the proposed algorithm is bounded by O(d,{varepsilon }^{-(4+delta )}) for any delta in (0,infty ) under suitable assumptions, where din {{mathbb {N}}} is the dimensionality of the problem and {varepsilon }in (0,infty ) is the prescribed accuracy. Moreover, the introduced class of numerical algorithms is also powerful for proving high-dimensional approximation capacities for deep neural networks.

Semiglobal Oblique Projection Exponential Dynamical Observers for Nonautonomous Semilinear Parabolic-Like Equations

The estimation of the full state of a nonautonomous semilinear parabolic equation is achieved by a Luenberger type dynamical observer. The estimation is derived from an output given by a finite number of average measurements of the state on small regions. The state estimate given by the observer converges exponentially to the real state, as time increases. The result is semiglobal in the sense that the error dynamics can be made stable for an arbitrary given initial condition, provided a large enough number of measurements, depending on the norm of the initial condition, is taken. The output injection operator is explicit and involves a suitable oblique projection. The results of numerical simulations are presented showing the exponential stability of the error dynamics.

Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation

In this paper, we propose L^2(J;H^1_0(Omega )) and L^2(J;L^2(Omega )) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.

Optimisation of the total population size with respect to the initial condition for semilinear parabolic equations: two-scale expansions and symmetrisations

In this article, we propose in-depth analysis and characterisation of the optimisers of the following optimisation problem: how to choose the initial condition u 0 in order to maximise the spatial integral at a given time of the solution of the semilinear equation u t −Δu = f(u), under L ∞ and L 1 constraints on u 0? Our contribution in the present paper is to give a characterisation of the behaviour of the optimiser when it does not saturate the L ∞ constraints, which is a key step in implementing efficient numerical algorithms. We give such a characterisation under mild regularity assumptions by proving that in that case can only take values in the ‘zone of concavity’ of f. This is done using two-scale asymptotic expansions. We then show how well-known isoperimetric inequalities yield a full characterisation of maximisers when f is convex. Finally, we provide several numerical simulations in one and two dimensions that illustrate and exemplify the fact that such characterisations significantly improve the computational time. All our theoretical results are in the one-dimensional case and we offer several comments about possible generalisations to other contexts, or obstructions that may prohibit doing so.

Inertial manifolds for a singularly non-autonomous semi-linear parabolic equations

This paper devotes to the existence of an N N -dimensional inertial manifold for a class of singularly, i.e. A ( t ) A(t) may degenerate to 0 0 at some time t t , non-autonomous parabolic equations ∂ t u + A ( t ) u = F ( t , u ) + g ( x , t ) , t > τ ; u | t = τ = u τ ( x ) , x ∈ Ω , \begin{equation*} \partial _{t}u+A(t)u=F(t,u)+g(x,t),\;t>\tau ;\; \; u|_{t=\tau }=u_{\tau }(x),\;x\in \Omega , \end{equation*} where A ( t ) ≥ 0 A(t)\geq 0 for any t ≥ τ t\geq \tau , and Ω ⊂ R d \Omega \subset \mathbb {R}^{d} is a bounded domain with smooth boundary. Since the operator A ( t ) A(t) may degenerate, a compatibility condition for the operator A ( t ) A(t) and the nonlinear term F ( t , u ) F(t,u) was proposed to construct the inertial manifolds.

Smooth invariant manifolds for a randomly perturbed non-autonomous coupled system and their approximations

We study long time dynamics of a randomly perturbed non-autonomous coupled system (x,y), whose x coordinate satisfies a semilinear parabolic equation with an additive noise, and y coordinate satisfies a differential equation whose solutions do not converge too rapidly. The noise is either the white noise induced by a Brownian motion W(t,ω) or a stationary process ζδ(θtω) whose integral is approximating W(t,ω). After addressing certain assumptions for such system, we show that for δ=0 (resp. δ>0) with respect to the noise W(t,ω) (resp. integral of ζδ(θtω)) there exists a invariant manifold which is exponentially attracting any other solution outside it. Also, as δ tends to 0, the invariant manifold and its derivative in y for the case δ>0 are approaching to those for δ=0.

Stationary approximations of inertial manifolds for stochastic retarded semilinear parabolic equations

A class of retarded semilinear parabolic equation driven by an additive Q-Wiener process is studied. We construct a process via the Wiener shift to provide stationary approximations of the noise, and show that solutions of stationary approximations converge pointwisely to solutions of the stochastic retarded semilinear parabolic equation. We also prove the existence of Ck,γ smooth inertial manifolds for the stationary approximations, and show that such inertial manifolds converge to the those of the stochastically perturbed retarded semilinear parabolic equation.

Carleman estimates and controllability results for fully discrete approximations of 1D parabolic equations

In this paper, we prove a Carleman estimate for fully discrete approximations of one-dimensional parabolic operators in which the discrete parameters h and △t are connected to the large Carleman parameter. We use this estimate to obtain relaxed observability inequalities which yield, by duality, controllability results for fully discrete linear and semilinear parabolic equations.

Posteriori Error bound For Fullydiscrete Semilinear Parabolic Integro-Differential equations

The main goal of this paper is to obtain error bounds for parabolic integro-differential equation. The derivation of these bounds is based elliptic and Ritz -Volterra reconstructions introduced by Makridakis and Nochetto 2003 and then extended to Ritz - Volterra reconstruction in case of integro-parabolic differential problems by Reddy and Sinha 2015. We proved optimal order bounds for certain classes semilinear parabolic intergro differential equations. The key points for these estimators can be reduced the numbers of iterations.

A generalized exponential stabilization for a class of semilinear parabolic equations: Linear boundary feedback control approach

This paper investigates the generalized exponential stabilization (GES) of a class of semilinear parabolic equations, which is a generalization of the result of exponential stabilization and has the character of accelerating decay process locally. From the point of admissibility of unbounded input operators of view, the well‐posedness of the open‐loop system is analyzed. Then, under some sufficient conditions expressed by linear matrix inequality, a linear boundary output feedback controller only using boundary measurement is proposed such that the closed‐loop system satisfies GES in absence of external disturbance and satisfies an H∞ performance index when disturbance exists. The well‐posedness of the resulting closed‐loop system is also given via semi‐group theory. Finally, we give numerical examples to verify the performance of the designed boundary controller.

Multiplicative Controllability of Semilinear Parabolic Equations with Neumann Boundary Conditions

Multiplicative Controllability of Semilinear Parabolic Equations with Neumann Boundary Conditions

Battery internal temperature estimation via a semilinear thermal PDE model

Accurate Lithium-ion (Li-ion) battery internal temperature information enables high-fidelity monitoring and safe operation in battery management systems, thus prevents thermal faults that could cause catastrophic failures. This paper proposes an online temperature estimation scheme for cylindrical Li-ion batteries based on a one-dimensional semilinear parabolic partial differential equation (PDE) model subject to in-domain and output uncertainties, using temperature measurements at the battery surface only. The thermal state observer design exploits PDE backstepping method, with a mild assumption on the Lipschitz continuity of the nonlinear heat generation rate. A sufficient condition on the Lipschitz constant to achieve exponential convergence is derived. Furthermore, when the thermal system uncertainties are present, an analytic bound on the temperature estimation error is formulated in the sense of spatial L2 norm, in terms of Lipschitz constant, design parameters, and bounds on system uncertainties. Simulation studies on various practical current profiles are demonstrated to illustrate the effectiveness of the proposed thermal estimation framework on a commercial cylindrical Li-ion battery cell.