Incomplete quenching in a system of heat equations coupled at the boundary

In this paper we find a possible continuation for quenching solutions to a system of heat equations coupled at the boundary condition. This system exhibits simultaneous and non-simultaneous quenching. For non-simultaneous quenching our continuation is a solution of a parabolic problem with Neumann boundary conditions. We also give some results for simultaneous quenching and present some numerical experiments that suggest that the approximations are not uniformly bounded in this case.     

Quenching rates for heat equations with coupled singular nonlinear boundary flux

We study finite time quenching for heat equations coupled via singular nonlinear bound-ary flux. A criterion is proposed to identify the simultaneous and non-simultaneous quenchings. In particular, three kinds of simultaneous quenching rates are obtained for different nonlinear exponent re-gions and appropriate initial data. This extends an original work by Pablo, Quir′os and Rossi for a heat system with coupled inner absorption terms subject to homogeneous Neumann boundary conditions.     

Non-simultaneous quenching for a slow diffusion system coupled at the boundary

This paper deals with the quenching behavior of positive solutions to the Newton filtration equations coupled with boundary singularities.We determine quenching rates for nonsimultaneous quenching at f...     

Non-simultaneous blowup in heat equations with nonstandard growth conditions

This paper cares about blowup solutions for a system of n-componential heat equations coupled via localized reactions and with variable exponents. The criteria for non-simultaneous and simultaneous blowup are established for radial solutions with or without assumptions on initial data, including the existence of non-simultaneous blowup for n components; any blowup must be simultaneous or non-simultaneous.     

Non-simultaneous quenching in a system of heat equations coupled at the boundary

We study the solutions of a parabolic system of heat equations coupled at the boundary through a nonlinear flux. We characterize in terms of the parameters involved when non-simultaneous quenching may appear. Moreover, if quenching is non-simultaneous we find the quenching rate, which surprisingly depends on the flux associated to the other component.     

Non-simultaneous quenching in a system of heat equations coupled at the boundary

We study the solutions of a parabolic system of heat equations coupled at the boundary through a nonlinear flux. We characterize in terms of the parameters involved when non-simultaneous quenching may appear. Moreover, if quenching is non-simultaneous we find the quenching rate, which surprisingly depends on the flux associated to the other component. Partially supported by project BFM2002-04572 (Spain). Partially supported by UBA grant EX046, CONICET and Fundación Antorchas (Argentina). Received: February 17, 2004; revised: July 5, 2004     

Quenching for a reaction–diffusion system with logarithmic singularity

In this paper, we study the quenching phenomenon for a reaction–diffusion system with singular logarithmic source terms and positive Dirichlet boundary conditions. Some sufficient conditions for quenching of the solutions in finite time are obtained, and the blow-up of time-derivatives at the quenching point is verified. Furthermore, under appropriate hypotheses, the non-simultaneous quenching of the system is proved, and the estimates of quenching rate is given.     

Numerical quenching of a system of equations coupled at the boundary

We study numerical approximations of solutions of the following system of heat equations, coupled at the boundary through a nonlinear flux: where p and q are parameters. We prove that the solutions of a semidiscretization in space quench in finite time. Moreover, we describe in terms of p and q the simultaneous versus non‐simultaneous quenching phenomena. We also find the numerical extinction sets. Finally, in order to obtain the correct quenching rates in the non‐simultaneous case we present some adaptive methods. Copyright © 2009 John Wiley & Sons, Ltd.     

Stagnation-point flow and heat transfer over an exponentially shrinking sheet

An analysis is carried out to investigate the stagnation-point flow and heat transfer over an exponentially shrinking sheet. Using the boundary layer approximation and a similarity transformation in exponential form, the governing mathematical equations are transformed into coupled, nonlinear ordinary differential equations which are then solved numerically by a shooting method with fourth order Runge-Kutta integration scheme. The analysis reveals that a solution exists only when the velocity ratio parameter satisfies the inequality −1.487068 ? c/a. Also, the numerical calculations exhibit the existence of dual solutions for the velocity and the temperature fields; and it is observed that their boundary layers are thinner for the first solution (in comparison with the second). Moreover, the heat transfer from the sheet increases with an increase in c/a for the first solution, while the heat transfer decreases with increasing c/a for the second solution, and ultimately heat absorption occurs.     

Non-simultaneous blow-up in coupled heat equations with multi-nonlinearities

This paper studies coupled heat equations with multi-nonlinearities of six nonlinear parameters. The critical blow-up exponent is established via a complete classification for all the six nonlinear parameters, where a precise analysis on the geometry of Ω and the absorption coefficients is given for the balanced interaction situation among the multi-nonlinearities. The main attention is contributed to non-simultaneous phenomena in the model to determine the necessary and sufficient conditions of non-simultaneous blow-up with suitable initial data, as well as the conditions under which any blow-up must be non-simultaneous. Finally, the model of the paper is characterized by a comparison with the known results for other models.     

Non-simultaneous versus simultaneous quenching in a coupled nonlinear parabolic system

This paper deals with finite-time quenching for the nonlinear parabolic system with coupled singular absorptions: u_t=Δu−v^−p, v_t=Δv−u^−q in Ω×(0,T) subject to positive Dirichlet boundary conditions, where p,q>0, Ω is a bounded domain in with smooth boundary. We obtain the sufficient conditions for global existence and finite-time quenching of solutions, and then determine the blow-up of time-derivatives and the quenching set for the quenching solutions. As the main results of the paper, a very clear picture is obtained for radial solutions with Ω=B_R: the quenching is simultaneous if p,q≥1, and non-simultaneous if p<1≤q or q<1≤p; if p,q<1 with , then both simultaneous and non-simultaneous quenching may happen, depending on the initial data. In determining the non-simultaneous quenching criteria of the paper, some new ideas have been introduced to deal with the coupled singular inner absorptions and inhomogeneous Dirichlet boundary value conditions, in addition to techniques frequently used in the literature.     

Non-simultaneous blow up for coupled nonlinear diffusion equations with absorptions

This paper deals with coupled nonlinear diffusion equations with absorptions. We characterize the range of parameters for which non-simultaneous blow up occurs. We establish the necessary and sufficient conditions for the occurrence of non-simultaneous blow up with proper initial data. Moreover, we obtain the optimal condition under which any blow up is non-simultaneous.     

Critical Quenching Exponents for Heat Equations Coupled with Nonlinear Boundary Flux

We discuss the quenching phenomena for a system of heat equations coupled with nonlinear boundary flux. We determine a critical value for the exponents in the boundary flux, such that only in the super critical case the simultaneous quenching can happen for any solution.     

Quenching phenomenon of a time‐fractional diffusion equation with singular source term

In this paper, a time‐fractional diffusion equation with singular source term is considered. The Caputo fractional derivative with order 0<α ?1 is applied to the temporal variable. Under specific initial and boundary conditions, we find that the time‐fractional diffusion equation presents quenching solution that is not globally well‐defined as time goes to infinity. The quenching time is estimated by using the eigenfunction of linear fractional diffusion equation. Moreover, by implementing a finite difference scheme, we give some numerical simulations to demonstrate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd.     

Existence of periodic solutions for a predator-prey system with sparse effect and functional response on time scales

In this paper, we systematically investigates the existence of periodic solutions of a predator-prey system with sparse effect and Beddington-DeAngelis or Holling III functional response on time scales. By using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of periodic solutions for the systems. Moreover, when the time scale T is chosen as R or Z, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the methods are unified to provide the existence of the desired solutions for the continuous differential equations and discrete difference equations.     

Weak Solutions to a Parabolic-Elliptic System of Chemotaxis

We study a parabolic-elliptic system of partial differential equations, which describes the chemotactic feature of slime molds. It is known that the blowup solution forms singularities such as delta functions, referred to as the collapses. Here, we study the case that the domain is a flat torus and show that the post-blowup continuation of the solution is possible only when those collapses are quantized with the mass 8π.     

Approximation of the solution of certain nonlinear ODEs with linear complexity

We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the “continuous” equation. Furthermore, we exhibit an algorithm computing an ε-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is linear in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.     

Existence result of second-order differential equations with integral boundary conditions at resonance

By using Mawhin's continuation theorem, some sufficient conditions for the existence of solution for a class of second-order differential equations with integral boundary conditions at resonance are established, which are complement of previously known results. The interesting point is that we shall deal with the case dimKerL=2, which will cause some difficulties in constructing the projector Q. Since all the existence results obtained in previous papers are for the case dimKerL=1, our work is new.     

Stochastic heat equation driven by fractional noise and local time

The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (0,1) in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos expansion for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.     

Periodic solutions for nonlinear nth order differential equations with delays

By applying the continuation theorem of coincidence degree theory, we establish the existence of 2π-periodic solutions for a class of nonlinear nth order differential equations with delays.