Instability of steady states for nonlinear wave and heat equations

We consider time-independent solutions of hyperbolic equations such as _∂ttu−Δu=f(x,u) where f is convex in u. We prove that linear instability with a positive eigenfunction implies nonlinear instability. In some cases the instability occurs as a blow up in finite time. We prove the same result for parabolic equations such as t_∂u−Δu=f(x,u). Then we treat several examples under very sharp conditions, including equations with potential terms and equations with supercritical nonlinearities.     

Note on a nonlinear diffusion system with convection

The critical exponents are established for a nonlinear diffusion system with convection, which are described clearly by the signs of two parameters solving the so-called characteristic algebraic system. It is proved that the convection plays an important role in determining the critical properties of solutions in the balance case. This greatly improves the authors’ previous paper for the same model.     

A note on the stability criterion for a class of nonlinear fractional differential systems

This paper is devoted to dealing with a flaw that existed in a recent paper (Zhou et al. 2014). We give a new proof of Th. 3.1 in Zhou et al. (2014), which is a correction of the original proof.     

A nonlinear diffusion system coupled via nonlinear boundary flux

In this paper, we deal with the global existence and nonexistence of solutions to a nonlinear diffusion system coupled via nonlinear boundary flux. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Wang et al. [Nonlinear Anal. 71 (2009) 2134-2140] and Li et al. [J. Math. Anal. Appl. 340 (2008) 876-883] to more general equations.     

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.     

Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

Influence of variable heat flux on natural convection along a corrugated wall in porous media

In this work the coupled non-linear partial differential equations, governing the free convection from a wavy vertical wall under a power law heat flux condition, are solved numerically. For both Darcy and Forchheimer extended non-Darcy models, a wavy to flat surface transformation is applied and the governing equations are reduced to boundary layer equations. A finite difference scheme based on the Keller Box approach has been used in conjunction with a block tri-diagonal solver for obtaining the solution. Detailed simulations are carried out to investigate the effect of varying parameters such as power law heat flux exponent m, wavelength–amplitude ratio a and the transformed Grashof number Gr′. Both surface undulations and inertial forces increase the temperature of the vertical surface while increasing m reduces it. The wavy pattern observed in surface temperature plots, become more prominent with increasing m or a but reduces as Gr′ increases.     

Natural convection boundary layer flow of fluid with temperature-dependent viscosity from a horizontal elliptical cylinder with constant surface heat flux

This study deals with the temperature-dependent viscosity effects on the natural convection boundary layer on a horizontal elliptical cylinder with constant surface heat flux. The mathematical problem is reduced to a pair of coupled partial differential equations for the temperature and the stream function, and the resulting nonlinear equations are solved numerically by cubic spline collocation method. Results for the heat transfer characteristics are presented as functions of eccentric angle for various values of viscosity variation parameters, Prandtl numbers and aspect ratios. Results show that an increase in the viscosity variation parameter tends to accelerate the fluid flow near the surface and increase the maximum velocity, thus decreasing the velocity boundary layer thickness. As the viscosity variation parameter is increased, the surface temperature tends to decrease, thus increasing the local Nusselt number. Moreover, the local Nusselt number of the elliptical cylinder increases as the Prandtl number of the fluid is increased.     

Blow-up rate for a nonlinear diffusion equation

In this work we study the blow-up rate for a nonlinear diffusion equation with an inner source and a nonlinear boundary flux, which is equivalent to a porous medium equation with convection. Depending upon the sign of a parameter included, the source can be positive or negative (absorption). By the scaling method, we obtain that the blow-up rate is independent of a negative source, while for the situation with a positive source, the blow-up rate is determined by the interaction between the inner source and the boundary flux. Comparing with the previous results for the porous medium model without convection, we observe that the gradient term included here does not affect the blow-up rates of solutions.     

Critical exponents for a fast diffusive polytropic filtration equation with nonlinear boundary flux

In this paper, we deal with a fast diffusive polytropic filtration equation

     

A note on flux limiting for diffusion discretizations

In this note, a limiting technique is presented to enforcemonotonicity for higher-order spatial diffusion discretizations.The aim is to avoid spurious oscillations and to improve thequalitative behaviour on coarse grids. The technique is relatedto known ones for convection equations, using limiters to boundthe numerical fluxes. Applications arise in pattern formationproblems for reaction–diffusion equations.     

A backward nonlinear heat equation: regularization with error estimates

In this article the authors consider a backward nonlinear heat equation. The uniqueness of the problem is proved and the problem is regularized by finite dimensional approximations. Error estimates in some particular cases are given.     

Natural convection flow from a horizontal circular cylinder with uniform heat flux in presence of heat generation

Natural convection boundary layer laminar flow from a horizontal circular cylinder with uniform heat flux in presence of heat generation has been investigated. The governing boundary layer equations are transformed into a non-dimensional form and the resulting non-linear systems of partial differential equations, which are solved numerically by two distinct methods namely: (i) implicit finite difference method together with the Keller-box scheme and (ii) perturbation solution technique. The results of the surface shear stress in terms of local skin-friction and the rate of heat transfer in terms of local Nusselt number, velocity distribution, velocity vectors, temperature distribution as well as streamlines, isotherms and isolines of pressure are shown by graphically for a selection of parameter set consisting of heat generation parameter.     

A note on generalized nonlinear diagonal dominance

In this paper, an open problem, proposed by A. Frommer, about nonlinear generalized diagonal dominance, is solved on some weak restriction, a counterexample is presented if such a restriction is omitted, and some new properties of nonlinear generalized diagonally dominant functions are investigated.     

A semilinear heat equation with a localized nonlinear source and non‐continuous initial data

This paper is devoted to the study of the Cauchy problem for a semilinear heat equation with nonlinear term presenting a nonlinear source centered in a closed region of the spatial domain Ω. We assume that is either a smooth bounded domain or the whole space , The initial data is assumed to belong to the Lebesgue space . Copyright © 2011 John Wiley & Sons, Ltd.     

Approximate solutions to one-dimensional nonlinear heat conduction problems with a given flux

One-dimensional (planar, cylindrically symmetric, and spherically symmetric) nonlinear heat conduction problems with the heat flux at the origin specified in the form of a power time dependence are considered. The initial temperature of the medium is assumed to be zero. Approximate solutions to the problems are obtained. The convergence of the resulting solutions is discussed.     

The space localization of unbounded boundary perturbations in nonlinear heat conduction with transfer

We consider the nonlinear parabolic equation u_t = (k(u)u_x)_x + b(u)_x, where u = u(x, t, x ε R¹, t > 0; k(u) ≥ 0, b(u) ≥ 0 are continuous functions as u ≥ 0, b (0) = 0; k, b > 0 as u > 0. At t = 0 nonnegative, continuous and bounded initial value is prescribed. The boundary condition u(0, t) = Ψ(t) is supposed to be unbounded as t → +∞. In this paper, sufficient conditions for space localization of unbounded boundary perturbations are found. For instance, we show that nonlinear equation u_t = (uⁿu_x)_x + (u^β)_x, n ≥ 0, β >; n + 1, exhibits the phenomenon of “inner boundedness,” for arbitrary unbounded boundary perturbations.     

Convergence to equilibrium for a parabolic-hyperbolic phase field model with Cattaneo heat flux law

In this paper we consider the well-posedness and the asymptotic behavior of solutions to the following parabolic-hyperbolic phase field system:

(0.1)