Semilinear Diffusion Equations Research Articles

Blow-up phenomena for positive solutions of semilinear diffusion equations in a half-space: the influence of the dispersion kernel

We consider the semilinear diffusion equation ∂ t u = Au + |u| α u in the half-space R N + := R N −1 × (0, +∞), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ′ , 0) = 0 for x ′ ∈ R N −1. We prove that if the symbol of the operator A is of order a|ξ| β near the origin ξ = 0, for some β ∈ (0, 2], then any positive solution of the semilinear diffusion equation blows up in finite time whenever 0 β/(N + 1). Notice that in the case of the half-space, the exponent β/(N + 1) is smaller than the so-called Fujita exponent β/N in R N. As a consequence we can also solve the blow-up issue for solutions of the above mentioned semilinear diffusion equation in the whole of R N , which are odd in the x N direction (and thus sign changing).

On inverse problem for linear and semilinear diffusion equation with Caputo–Fabrizio derivative

We investigate a fractional time diffusion equation with Caputo–Fabrizio derivative instead of classical derivative. We set up the existence, uniqueness, and regularity of the mild solution and then consider the above problem in some various cases including linear and nonlinear source function. Thanks to applying some Sobolev embeddings, we derive the regularity of the mild solution for linear case. For nonlinear case, we apply the fixed point theory to obtain the existence of the mild solution.

Convergence analysis of a numerical scheme for a tumour growth model

Abstract A one–spatial dimensional tumour growth model Breward et al. ( 2001, 2002, 2003) that consists of three dependent variables of space and time is considered. These variables are volume fraction of tumour cells, velocity of tumour cells, and nutrient concentration. The model variables satisfy a coupled system of semilinear advection equation (hyperbolic), simplified linear Stokes equation (elliptic), and semilinear diffusion equation (parabolic) with appropriate conditions on the time–dependent boundary, which is governed by an ordinary differential equation. A reformulation of the model defined in a larger, fixed time–space domain is employed to overcome theoretical difficulties related to the time–dependent boundary. This reformulation reduces the complexity of the model by removing the need to explicitly track the time–dependent boundary. A numerical scheme that employs a finite volume method for the hyperbolic equation, a finite element method for the elliptic equation, and a backward Euler in time–mass lumped finite element in space method for the parabolic equation is developed. We establish the existence of a time interval $(0,T_{\ast })$ over which a convergent subsequence of the numerical approximations can be extracted using compactness techniques. The limit of any such convergent subsequence is proved to be a weak solution of the continuous model in an appropriate sense, which is called a threshold solution. Numerical tests and justifications that support the theoretical findings conclude the paper.

Analysis of Caputo-Fabrizio fractional order semi-linear parabolic equations via effective amalgamated technique

The significance of semi-linear parabolic equations in various fields of physics and chemistry is perpetual. Literature is enriched with the modeling and numerical investigations of their various paradigms. In this paper, a class of semi-linear diffusion equations is considered as prototypical semi-linear parabolic equation. The equations are reformulated to fractional order derivative by applying Caputo-Fabrizio time fractional derivative (CFTFD). Moreover, an amalgamated technique, that is, a semi-analytical technique is also established, which is combination of Laplace transform and Picard’s iterative method (LTPIM). Specifically, it is designed to effectively simulate the governing semi-linear diffusion equations. In addition, the stability analysis of this amalgamated technique is also carried out through comparison with Banach fixed point theorem and H-stable mapping. The obtained results are illustrated graphically and in tabulated form, which evidently validates the proficiency of this technique for semi-linear parabolic equations.

Numerical study for time fractional stochastic semi linear advection diffusion equations

In this work, a stochastic fractional advection diffusion model with multiplicative noise is studied numerically. The Galerkin finite element method in space and finite difference in time are used, where the fractional derivative is in Caputo sense. The error analysis is investigated via Galerkin finite element method. In terms of the Mittag Leffler function, the mild solution is obtained. For the error estimates, the strong convergence for the semi and fully discrete schemes are proved in a semigroup structure. Finally, two numerical examples are given to confirm the theoretical results.

Remarks on global solutions for the semilinear diffusion equation in the de Sitter spacetime

The Cauchy problem for the semilinear diffusion equation is considered in the de Sitter spacetime with the spatial zero-curvature. Global solutions and their asymptotic behaviors for small initial data are obtained for positive and negative Hubble constants. The effects of the spatial expansion and contraction are studied on the problem.

Asymptotic behaviors of global solutions for a semilinear diffusion equation in the de Sitter spacetime

A semilinear diffusion equation is considered in the de Sitter spacetime with the spatially flat curvature. Global solutions for small initial data and their asymptotic behaviors in high order are obtained. Some effects of the spatial expansion and contraction are studied through the problem.

Blow-up of sign-changing solutions for a one-dimensional nonlinear diffusion equation

This paper is concerned with a spatially inhomogeneous semilinear diffusion equation on a bounded interval under the Neumann or Dirichlet boundary conditions. Assuming that the nonlinearity satisfies a rather general growth condition, we consider the blow-up and global existence of sign-changing solutions. It is shown that for some nonnegative integer k depending on the linearized operator at a trivial solution, the solution blows up in finite time if an initial value changes its sign at most k times, whereas there exist stationary solutions with more than k zeros. The proof is based on the intersection number principle combined with a topological method.

Sharp boundary behaviour of solutions to semilinear nonlocal elliptic equations

We investigate quantitative properties of nonnegative solutions $$u(x)\ge 0$$ to the semilinear diffusion equation $$\mathcal {L}u= f(u)$$ , posed in a bounded domain $$\Omega \subset \mathbb {R}^N$$ with appropriate homogeneous Dirichlet or outer boundary conditions. The operator $$\mathcal {L}$$ may belong to a quite general class of linear operators that include the standard Laplacian, the two most common definitions of the fractional Laplacian $$(-\Delta )^s$$ ( $$0<s<1$$ ) in a bounded domain with zero Dirichlet conditions, and a number of other nonlocal versions. The nonlinearity f is increasing and looks like a power function $$f(u)\sim u^p$$ , with $$p\le 1$$ . The aim of this paper is to show sharp quantitative boundary estimates based on a new iteration process. We also prove that, in the interior, solutions are Hölder continuous and even classical (when the operator allows for it). In addition, we get Hölder continuity up to the boundary. Particularly interesting is the behaviour of solution when the number $$\frac{2s}{1-p}$$ goes below the exponent $$\gamma \in (0,1]$$ corresponding to the Hölder regularity of the first eigenfunction $$\mathcal {L}\Phi _1=\lambda _1 \Phi _1$$ . Indeed a change of boundary regularity happens in the different regimes $$\frac{2s}{1-p} \gtreqqless \gamma $$ , and in particular a logarithmic correction appears in the “critical” case $$\frac{2s}{1-p} = \gamma $$ . For instance, in the case of the spectral fractional Laplacian, this surprising boundary behaviour appears in the range $$0<s\le (1-p)/2$$ .

Parallel-In-Space-Time, Adaptive Finite Element Framework for Nonlinear Parabolic Equations

We present an adaptive methodology for the solution of (linear and) nonlinear time dependent problems that is especially tailored for massively parallel computations. The basic concept is to solve for large blocks of space-time unknowns instead of marching sequentially in time. The methodology is a combination of a computationally efficient implementation of a parallel-in-space-time finite element solver coupled with a posteriori space-time error estimates and a parallel mesh generator. While we focus on spatial adaptivity in this work, the methodology enables simultaneous adaptivity in both space and time domains. We explore this basic concept in the context of a variety of time steppers including $\Theta$-schemes and backward difference formulas. We specifically illustrate this framework with applications involving time dependent linear, quasi-linear, and semilinear diffusion equations. We focus on investigating how the coupled space-time refinement indicators for this class of problems affect spatial adaptivity. Finally, we show good scaling behavior up to 150,000 processors on the NCSA Blue Waters machine. This conceptually simple methodology enables scaling on next generation multicore machines by simultaneously solving for a large number of timesteps, and reducing computational overhead by locally refining spatial blocks that can track localized features. This methodology also opens up the possibility of efficiently incorporating adjoint equations for error estimators and inverse design problems, since blocks of space-time are simultaneously solved and stored in memory.

Blowup of Volterra Integro-Differential Equations and Applications to Semi-Linear Volterra Diffusion Equations

AbstractIn this paper, we discuss the blowup of Volterra integro-differential equations (VIDEs) with a dissipative linear term. To overcome the fluctuation of solutions, we establish a Razumikhin-type theorem to verify the unboundedness of solutions. We also introduce leaving-times and arriving-times for the estimation of the spending-times of solutions to ∞. Based on these two typical techniques, the blowup and global existence of solutions to VIDEs with local and global integrable kernels are presented. As applications, the critical exponents of semi-linear Volterra diffusion equations (SLVDEs) on bounded domains with constant kernel are generalized to SLVDEs on bounded domains and ℝN with some local integrable kernels. Moreover, the critical exponents of SLVDEs on both bounded domains and the unbounded domain ℝN are investigated for global integrable kernels.

Option prices under liquidity risk as weak solutions of semilinear diffusion equations

Prices of financial options in a market with liquidity risk are shown to be weak solutions of a class of semilinear parabolic partial differential equations with nonnegative characteristic form. We prove the existence and uniqueness of such solutions, and then show the solutions correspond to option prices as defined in terms of replication in a probabilistic setup. We obtain an asymptotic representation of the price and the hedging strategy as a liquidity parameter converges to zero.

Moving Finite Element Methods for a System of Semi-Linear Fractional Diffusion Equations

AbstractThis paper studies a system of semi-linear fractional diffusion equations which arise in competitive predator-prey models by replacing the second-order derivatives in the spatial variables with fractional derivatives of order less than two. Moving finite element methods are proposed to solve the system of fractional diffusion equations and the convergence rates of the methods are proved. Numerical examples are carried out to confirm the theoretical findings. Some applications in anomalous diffusive Lotka-Volterra and Michaelis-Menten-Holling predator-prey models are studied.

Два формулювання узагальненої задачі Коші для півлінійного рівняння дифузії з дробовою похідною за часом

Different equivalent definitions of the Cauchy problem for semi-linear diffusion equation with fractional derivative with respect to time and with the generalized function in the initial condition are offered. The existence and uniqueness theorem and the representation of the solution of such problem for linear homogeneous diffusion equation with fractional derivative with respect to time are obtained.

Robust sampled-data control of a class of semilinear parabolic systems

We develop sampled-data controllers for parabolic systems governed by uncertain semilinear diffusion equations with distributed control on a finite interval. Such systems are stabilizable by linear infinite-dimensional state-feedback controllers. For a realistic design, finite-dimensional realizations can be applied leading to local stability results. Here we suggest a sampled-data controller design, where the sampled-data (in time) measurements of the state are taken in a finite number of fixed sampling points in the spatial domain. It is assumed that the sampling intervals in time and in space are bounded. Our sampled-data static output feedback enters the equation through a finite number of shape functions (which are localized in the space) multiplied by the corresponding state measurements. It is piecewise-constant in time and it may possess an additional time-delay. The suggested controller can be implemented by a finite number of stationary sensors (providing discrete state measurements) and actuators and by zero-order hold devices. A direct Lyapunov method for the stability analysis of the resulting closed-loop system is developed, which is based on the application of Wirtinger’s and Halanay’s inequalities. Sufficient conditions for the exponential stabilization are derived in terms of Linear Matrix Inequalities (LMIs). By solving these LMIs, upper bounds on the sampling intervals that preserve the exponential stability and on the resulting decay rate can be found. The dual problem of observer design under sampled-data measurements is formulated, where the same LMIs can be used to verify the exponential stability of the error dynamics.

Long time asymptotics of sub-threshold solutions of a semilinear Cauchy problem

We show that the solution of the semilinear heat equation u t = uxx +u p (with x ∈ R, p > 3, and nonnegative Cauchy data) behaves for large t like the solution of the corresponding linear problem plus a small correction of order t −1/2−c ,w herec := 1/2, if p 4, and c = (p −3)/2, if 3< p < 4. The result is known in special cases like small initial data. We prove it here for positive sub-threshold initial data satisfying some assumptions. Part of our results are contained in the recent work (10), but the motivation of this paper is to provide a new method leading to somewhat more general space-time estimates. 1. Main result We consider the long time asymptotical behaviour of the solution of the classi- cal semilinear diffusion equation and the associated Cauchy problem for the unknown function u = u(x,t), x ∈ R, t 0:

Enhanced Group Analysis and Exact Solutions of Variable Coefficient Semilinear Diffusion Equations with a Power Source

A new approach to group classification problems and more general investigations on transformational properties of classes of differential equations is proposed. It is based on mappings between classes of differential equations, generated by families of point transformations. A class of variable coefficient (1+1)-dimensional semilinear reaction–diffusion equations of the general form f(x)u t =(g(x)u x ) x +h(x)u m (m≠0,1) is studied from the symmetry point of view in the framework of the approach proposed. The singular subclass of the equations with m=2 is singled out. The group classifications of the entire class, the singular subclass and their images are performed with respect to both the corresponding (generalized extended) equivalence groups and all point transformations. The set of admissible transformations of the imaged class is exhaustively described in the general case m≠2. The procedure of classification of nonclassical symmetries, which involves mappings between classes of differential equations, is discussed. Wide families of new exact solutions are also constructed for equations from the classes under consideration by the classical method of Lie reductions and by generation of new solutions from known ones for other equations with point transformations of different kinds (such as additional equivalence transformations and mappings between classes of equations).

Asymptotical behaviour of a class of semilinear diffusion equations

We improve the existing results of the long time asymptotical behaviour of some basic semilinear diffusion equations (with space variable in the whole space \(\mathbb{R}\)). Our method is elementary: it is based on explicit calculations, weighted sup–norm estimates and a fixed point argument.

Blow-up profile of a solution for a nonlinear heat equation with small diffusion

This paper is concerned with positive solutions of semilinear diffusion equations ut=ε2∆u+up in Ω with small diffusion under the Neumann boundary condition, where p>1 is a constant and Ω is a bounded domain in RN with C2 boundary. For the ordinary differential equation ut=up, the solution u0 with positive initial data u0∊C(Ω¯) has a blow-up set S0={x∊Ω¯|u0(x)=maxy∊Ω¯u0(y)} and a blowup profile u*0(x)=(u0(x)-(p-1)-(maxy∊Ω¯u0(y))-(p-1))-1/(p-1) outside the blow-up set S0. For the diffusion equation ut=ε2∆u+up in Ω under the boundary condition ∂u/∂v=0 on ∂Ω, it is shown that if a positive function u0∊C2(Ω¯) satisfies ∂u0/∂v=0 on ∂Ω, then the blow-up profile u*ε(x) of the solution uε with initial data u0 approaches u*0(x) uniformly on compact sets of Ω¯∖S0 as ε→+0.

Variable instability of a constant blow-up solution in a nonlinear heat equation

This paper is concerned with positive solutions of the semilinear diffusion equation ut=∆u+up in Ω under the Neumann boundary condition, where p>1 is a constant and Ω is a bounded domain in RN with C2 boundary. This equation has the constant solution (p-1)-1/(p-1)(T0-t)-1/(p-1)(0≤t 0. It is shown that for any e>0 and open cone Γ in {f∊C(Ω¯)|f(x)>0}, there exists a positive function u0(x) in Ω¯ with ∂u0/∂v=0 on ∂Ω and ||u0(x)-(p-1)-1/(p-1)T0-1/(p-1)||C2(Ω¯)<e such that the blow-up time of the solution u(x,t) with initial data u(x,0)=u0(x) is larger than T0 and the function u(x,T0) belongs to the cone Γ. A theorem on the blow-up profile is also given.