Semilinear Parabolic Systems Research Articles

Construction and stability of type I blowup solutions for non-variational semilinear parabolic systems

AbstractIn this note, we consider the semilinear heat system\partial_{t}u=\Delta u+f(v),\quad\partial_{t}v=\mu\Delta v+g(u),\quad\mu>0,where the nonlinearity has no gradient structure taking of the particular formf(v)=v\lvert v\rvert^{p-1}\quad\text{and}\quad g(u)=u\lvert u\rvert^{q-1}\quad% \text{with }p,q>1,orf(v)=e^{pv}\quad\text{and}\quad g(u)=e^{qu}\quad\text{with }p,q>0.We exhibit type I blowup solutions for this system and give a precise description of its blowup profiles. The method relies on a two-step procedure: the reduction of the problem to a finite-dimensional one via a spectral analysis, and then solving the finite-dimensional problem by a classical topological argument based on index theory. As a consequence of our technique, the constructed solutions are stable under a small perturbation of initial data. The results and the main arguments presented in this note can be found in our papers [T.-E. Ghoul, V. T. Nguyen and H. Zaag, Construction and stability of blowup solutions for a non-variational semilinear parabolic system, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 2018, 6, 1577–1630] and [M. A. Herrero and J. J. L. Velázquez, Generic behaviour of one-dimensional blow up patterns, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 19 1992, 3, 381–450].

Regional Observability with Constraints on the State of Semilinear Parabolic Systems

Regional Observability with Constraints on the State of Semilinear Parabolic Systems

Fuzzy Stabilization Design for Semilinear Parabolic PDE Systems With Mobile Actuators and Sensors

This paper deals with a fuzzy stabilization design problem for a class of semilinear parabolic partial differential equation (PDE) systems using mobile actuators and sensors. Initially, a Takagi–Sugeno (T–S) fuzzy PDE model is employed to accurately represent the semilinear parabolic PDE system. Subsequently, based on the T–S fuzzy model, a stabilization scheme containing the fuzzy controllers and the guidance of mobile actuator/sensor pairs is proposed, where the spatial domain is decomposed into multiple subdomains according to the number of actuator/sensor pairs and each actuator/sensor pair is capable of moving within the respective subdomain. Then, by a Lyapunov direct technique, an integrated design of fuzzy controllers plus mobile actuator/sensor guidance laws is developed in the form of bilinear matrix inequalities (BMIs), such that the resulting closed-loop system is exponentially stable and the mobile actuator/sensor guidance can enhance the transient performance of the closed-loop system. Furthermore, an iterative algorithm based on linear matrix inequalities is proposed to solve the BMIs. Finally, two examples are given to illustrate the effectiveness of the proposed method.

High-accuracy numerical methods for a parabolic system in air pollution modeling

We present two approaches for enhancing the accuracy of the second-order finite difference approximations of two-dimensional semilinear parabolic systems. These are the fourth-order compact difference scheme and the fourth-order scheme based on Richardson extrapolation. Our interest is concentrated on a system of ten parabolic partial differential equations in air pollution modeling. We analyze numerical experiments to compare the two approaches with respect to accuracy, computational complexity, nonnegativity preserving, etc. The sixth-order approximation based on the fourth-order compact difference scheme combined with Richardson extrapolation is also discussed numerically.

Differential Harnack Estimates for a Semilinear Parabolic System

In this paper, we prove differential Harnack inequalities for positive solutions of a semilinear parabolic system on hyperbolic space. We use the inequalities to construct classical Harnack estimates by integrating along space-time.

Simultaneous Quenching for Semilinear Parabolic System with Localized Sources in a Square Domain

This paper studies quenching properties of solutions of a semilinear parabolic system with localized reaction sources in a square domain. The system has the homogeneous Dirichlet boundary condition and null initial condition. We prove that solutions quench simultaneously, and compute approximated critical values of the system using a numerical method.

Observer Design For Multidimensional Parabolic Systems

Abstract In this paper we present some observer designs for infinite dimensional parabolic systems, defined in multidimensional space domains. We consider two observation scenarios. (i) When the state is measured in an open (possibly small) subdomain of the space domain we obtain an exponentially convergent observer with gain for a semilinear parabolic system in arbitrary domains. (ii) When the state is measured along a line orthogonal to the sides of a rectangle in ℝ2 we derive an observer for a linear parabolic system. The mathematical setting relies upon the semigroup theory for n-dimensional autonomous systems as well as on Fourier analysis.

Fujita-type theorems for a class of coupled semilinear parabolic systems with gradient terms

This paper concerns the asymptotic behavior of the solution to a class of coupled semilinear parabolic systems with gradient terms. The Fujita-type blow-up theorems are established and the critical Fujita curve is determined not only by the behavior of the coefficients of the gradient term and the source terms at infinity, but also by the spacial dimension.

Strong and weak stabilization of semi-linear parabolic systems

Strong and weak stabilization of semi-linear parabolic systems

Backward Parabolicity, Cross-Diffusion and Turing Instability

We show that the ill-posedness observed in backward parabolic equation, or cross-diffusion systems, can be interpreted as a limiting Turing instability for a corresponding semi-linear parabolic system. Our analysis is based on the, now well established, derivation of nonlinear parabolic and cross-diffusion systems from semi-linear reaction–diffusion systems with fast reaction rates. We illustrate our observation with two generic examples for $$2\times 2$$ and $$4\times 4$$ reaction–diffusion systems. For these examples, we prove that backward parabolicity in cross-diffusion systems is equivalent to Turing instability for fast reaction rates. In one dimension, the Turing patterns are periodic solutions which have frequencies which increase with the reaction rate. Furthermore, in some specific cases, the structure of the equations at hand involves classical entropy/Lyapunov functions which lead to a priori estimates allowing to pass rigorously to the fast reaction limit in the absence of Turing instabilities.

Construction and stability of blowup solutions for a non-variational semilinear parabolic system

We consider the following parabolic system whose nonlinearity has no gradient structure:{∂tu=Δu+|v|p−1v,∂tv=μΔv+|u|q−1u,u(⋅,0)=u0,v(⋅,0)=v0, in the whole space RN, where p,q>1 and μ>0. We show the existence of initial data such that the corresponding solution to this system blows up in finite time T(u0,v0) simultaneously in u and v only at one blowup point a, according to the following asymptotic dynamics:{u(x,t)∼Γ[(T−t)(1+b|x−a|2(T−t)|log⁡(T−t)|)]−(p+1)pq−1,v(x,t)∼γ[(T−t)(1+b|x−a|2(T−t)|log⁡(T−t)|)]−(q+1)pq−1, with b=b(p,q,μ)>0 and (Γ,γ)=(Γ(p,q),γ(p,q)). The construction relies on the reduction of the problem to a finite dimensional one and a topological argument based on the index theory to conclude. Two major difficulties arise in the proof: the linearized operator around the profile is not self-adjoint even in the case μ=1; and the fact that the case μ≠1 breaks any symmetry in the problem. In the last section, through a geometrical interpretation of quantities of blowup parameters whose dimension is equal to the dimension of the finite dimensional problem, we are able to show the stability of these blowup behaviors with respect to perturbations in initial data.

Output stabilization of semilinear parabolic systems with bounded feedback

In this paper, we will study the output feedback stabilization of infinite-semilinear parabolic systems evolving on a spatial domain Ω and in a subregion ω of Ω (interior to Ω or on its boundary ∂Ω). We consider the condition of admissibility and the decomposition methods technique of the state space via the spectral properties of the system. Then we apply this approach to a regional exponential stabilization problem using bounded feedback. Applications are presented.

On a free boundary problem for semilinear parabolic system

On a free boundary problem for semilinear parabolic system

Exponential Pointwise Stabilization of Semilinear Parabolic Distributed Parameter Systems via the Takagi–Sugeno Fuzzy PDE Model

This paper deals with the problem of exponential stabilization for nonlinear parabolic distributed parameter systems using the Takagi–Sugeno (T–S) fuzzy partial differential equation (PDE) model, where a finite number of actuators are active only at some specified points of the spatial domain (these actuators are referred to as pointwise actuators). Three cases of state feedback are respectively considered in this study as follows: full state feedback, piecewise state feedback, and collocated pointwise state feedback. It is initially assumed that a T–S fuzzy PDE model obtained via the sector nonlinearity approach is employed to accurately represent the semilinear parabolic PDE system. Based on the obtained T–S fuzzy PDE model, Lyapunov-based design methodologies of fuzzy feedback control laws are subsequently derived for the above three state feedback cases by using the vector-valued Wirtinger's inequality to guarantee locally exponential pointwise stabilization of the semilinear PDE system, and presented in terms of standard linear matrix inequalities (LMIs). Moreover, the favorable property offered by sharing all the same premises in the T–S fuzzy PDE models and fuzzy controllers is not applicable for the case of collocated pointwise state feedback. A parameterized LMI is introduced for this case to enhance the stabilization ability of the fuzzy controller. Finally, the merit and effectiveness of the proposed design methods are demonstrated by numerical simulation results of two examples.

Regional boundary controllability of semi-linear parabolic systems with state constraints

This work focuses on the controllability of semi-linear parabolic systems with state constraints. Sub-differential techniques are used to compute the control u that steers the system (S) from the initial state y0 to a final one between two prescribed functions, only on a boundary subregion ᴦ of the system evolution domain Ω.

Regional boundary controllability of semi-linear parabolic systems with state constraints

Regional boundary controllability of semi-linear parabolic systems with state constraints

A Liouville-type theorem for cooperative parabolic systems

We prove Liouville-type theorem for semilinear parabolic system of the form $u_t-\Delta u =a_{11}u^{p}+a_{12} u^rv^{s+1}$, $v_t-\Delta v =a_{21} u^{r+1}v^{s}+a_{22}v^{p}$ where $r, s>0$, $p=r+s+1$. The real matrix $A=(a_{ij})$ satisfies conditions $ a_{12}, a_{21}\geq 0$ and $a_{11}, a_{22}>0$. This paper is a continuation of Phan-Souplet (Math. Ann., 366,1561-1585,2016) where the authors considered the special case $s=r$ for the system of $m$ components. Our tool for the proof of Liouville-type theorem is a refinement of Phan-Souplet, which is based on Gidas-Spruck (Commun. Pure Appl.Math. 34,525–598 1981) and Bidaut-Véron (Équations aux dérivées partielles et applications. Elsevier, Paris, pp 189–198,1998).

Asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with gradient terms

Asymptotic behavior of solutions to a class of coupled semilinear parabolic systems with gradient terms

Guaranteed cost fuzzy state observer design for semilinear parabolic PDE systems under pointwise measurements

This paper studies the guaranteed cost fuzzy state observer (GCFSO) design via pointwise measurement sensors for a class of distributed parameter systems described by semilinear parabolic partial differential equations (PDEs). Initially, a Takagi–Sugeno (T–S) fuzzy model is employed to accurately represent the original PDE system in a local region. Then, based on the T–S fuzzy model, a fuzzy state observer is constructed for the state estimation. By augmenting the state estimation error system with the fuzzy PDE system and utilizing Lyapunov technique and Wirtinger’s inequality, a fuzzy state observer design with a guaranteed cost is developed in terms of linear matrix inequalities (LMIs). The resulting fuzzy observer can exponentially stabilize the augmented system while providing an upper bound on the cost function of state estimation error. Moreover, a suboptimal GCFSO design problem is also addressed to make the upper bound as small as possible. Finally, the numerical simulation results on two examples demonstrate the effectiveness of the proposed method.

Construction of type II blow-up solutions for a semilinear parabolic system with higher dimension

This paper is concerned with blow-up solutions for a semilinear parabolic system with a power type nonlinearity. Non self-similar blow-up solution is constructed by the matched asymptotic expansions. One component of this solution converges to the singular steady state, and another component converges to zero in self-similar variables.