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## Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients

### Journal of Nonlinear Science (2009-06-01) 19: 301-339 , June 01, 2009

We rigorously prove results on spiky patterns for the Gierer–Meinhardt system (Kybernetik (Berlin) 12:30–39, 1972) with a jump discontinuity in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini, and Sherratt (Math. Comput. Model. 17:29–34, 1993a; Bull. Math. Biol. 55:365–384, 1993b; IMA J. Math. Appl. Med. Biol. 9:197–213, 1992).

Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular, we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity. This *localization principle* for a spike is a *new effect* which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable.

Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a *spike near the jump discontinuity* is the second *new effect* presented in this paper.

To derive these new effects in a mathematically rigorous way, we use analytical tools like Liapunov–Schmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work (J. Nonlinear Sci. 11:415–458, 2001).

Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.

## Global Weak Solutions of 3D Compressible Nematic Liquid Crystal Flows with Discontinuous Initial Data and Vacuum

### Acta Applicandae Mathematicae (2016-04-01) 142: 149-171 , April 01, 2016

In this paper, we study the global existence of weak solutions to the Cauchy problem of the three-dimensional equations for compressible isentropic nematic liquid crystal flows subject to discontinuous initial data. It is assumed here that the initial energy is suitably small in *L*^{2}, and the initial density, the gradients of initial velocity/liquid crystal director field are bounded in *L*^{∞}, *L*^{2} and *H*^{1}, respectively. This particularly implies that the initial data may contain vacuum states and the oscillations of solutions could be arbitrarily large. As a byproduct, we also prove the global existence of smooth solutions with strictly positive density and small initial energy.

## Decay estimates for “anisotropic” viscous Hamilton-Jacobi equations in ℝ N

### Nonlinear Evolution Equations and Related Topics (2004-01-01): 27-37 , January 01, 2004

The large time behaviour of the *L*^{q}-norm of nonnegative solutions to the “anisotropic” viscous Hamilton-Jacobi equation
$$
{u_{t}} - \Delta u + {\sum\limits_{{i = 1}}^{m} {|{u_{{xi}}}|} ^{{Pi}}} = 0 in {\mathbb{R}_{ + }} x {\mathbb{R}^{N}},
$$
is studied for *q* = 1 and *q* = ∞, where *m* ∈ {1,...,*N*} and *p*_{i} for *i* ∈ {1,...,*m*}. The limit of the*L*^{1}-norm is identified, and temporal decay estimates for the *L*^{∞}-norm are obtained, according to the values of the *p*_{i}’s. The main tool in our approach is the derivation of L^{∞}-decay estimates for
$$
\nabla ({u^{\alpha }}),\alpha \in (0,1]
$$
, by a Bernstein technique inspired by the ones developed by Bénilan for the porous medium equation.

## On the long time behaviour of solutions to dissipative wave equations in $$\mathbb{R}^{2} $$

### Nonlinear Differential Equations and Applications NoDEA (2006-07-01) 13: 193-204 , July 01, 2006

### Abstract.

In this paper we present a parabolic approach to studying the diffusive long time behaviour of solutions to the Cauchy problem:
1
$$ \left\{ \begin{aligned} & u_{{ t t }} + u_{t} - \Delta u = 0,x \in R^{N} ,t \ > 0,\\ & u(\cdot, 0) = u_{0} ,x \in R^{N} ,\\ & u(\cdot, 0) = u_{1} ,x \in R^{N} x; \end{aligned} \right. $$
where *u*_{0} and *u*_{1} satisfy suitable assumptions.

After an appropriate scaling we obtain the convergence to a stationary solutio n in *L*^{q} norm (1 ≤ *q* < ∞).

## Large-time behavior of the strong solution to nonhomogeneous incompressible MHD system with general initial data

### Boundary Value Problems (2015-11-17) 2015: 1-13 , November 17, 2015

This paper investigates the large-time behavior of strong solutions to the nonhomogeneous incompressible magnetohydrodynamic equations on a bounded domain in $\mathbb{R}^{2}$ . Based on uniform estimates, we prove that the velocity, the magnetic field, and their derivatives converge to zero in $L^{2}$ norm as time goes to infinity without any additional assumption on the initial data and external force by a pure energy method.

## Threshold Behavior and Non-quasiconvergent Solutions with Localized Initial Data for Bistable Reaction–Diffusion Equations

### Journal of Dynamics and Differential Equations (2016-09-01) 28: 605-625 , September 01, 2016

We consider bounded solutions of the semilinear heat equation $$u_t=u_{xx}+f(u)$$ on $$R$$ , where $$f$$ is of the unbalanced bistable type. We examine the $$\omega $$ -limit sets of bounded solutions with respect to the locally uniform convergence. Our goal is to show that even for solutions whose initial data vanish at $$x=\pm \infty $$ , the $$\omega $$ -limit sets may contain functions which are not steady states. Previously, such examples were known for balanced bistable nonlinearities. The novelty of the present result is that it applies to a robust class of nonlinearities. Our proof is based on an analysis of threshold solutions for ordered families of initial data whose limits at infinity are not necessarily zeros of $$f$$ .

## Exponential Attractor for a Nonlinear Boussinesq Equation

### Acta Mathematicae Applicatae Sinica (2006-07-01) 22: 443-450 , July 01, 2006

###
*Abstract*

This paper is devoted to prove the existence of an exponential attractor for the semiflow generated by a nonlinear Boussinesq equation. We formulate the Boussinesq equation as an abstract equation in the Hilbert space $$ H^{2}_{0} {\left( {0,1} \right)} \times L^{2} {\left( {0,1} \right)} $$ . The main step in this research is to show that there exists an absorbing set for the solution semiflow in the Hilbert space $$ H^{3}_{0} {\left( {0,1} \right)} \times H^{1}_{0} {\left( {0,1} \right)} $$ .

## On a Class of Nonlinear Viscoelastic Kirchhoff Plates: Well-Posedness and General Decay Rates

### Applied Mathematics & Optimization (2016-02-01) 73: 165-194 , February 01, 2016

This paper is concerned with well-posedness and energy decay rates to a class of nonlinear viscoelastic Kirchhoff plates. The problem corresponds to a class of fourth order viscoelastic equations of $$p$$ -Laplacian type which is not locally Lipschitz. The only damping effect is given by the memory component. We show that no additional damping is needed to obtain uniqueness in the presence of rotational forces. Then, we show that the general rates of energy decay are similar to ones given by the memory kernel, but generally not with the same speed, mainly when we consider the nonlinear problem with large initial data.

## Asymptotic behaviour of hessian equation with boundary blow up

### Acta Mathematicae Applicatae Sinica, English Series (2017-07-01) 33: 575-586 , July 01, 2017

We consider the boundary blow up problem for *k*-hessian equation with nonlinearities of power and of exponential type, and prove their existence, uniqueness and asymptotic behaviour. Moreover we also show that their perturbed problem has a unique positive solution, which satisfies some asymptotic behaviors to unperturbed problems under appropriate structure hypotheses for perturbed terms.

## On a Convection Diffusion Equation with Absorption Term

### Bulletin of the Malaysian Mathematical Sciences Society (2017-04-01) 40: 523-544 , April 01, 2017

The paper studies the posedness of the convection diffusion equation $$\begin{aligned} u_{t}=\text {div}\left( \left| \nabla u^{m}\right| ^{p-2}\nabla u^{m}\right) +\sum _{i=1}^{N}\frac{\partial b_{i}\left( u^{m}\right) }{\partial x_{i}}-u^{mr}. \end{aligned}$$ with homogeneous boundary condition and with the initial value $$u_0(x)\in L^{q-1+\frac{1}{m}}(\Omega )$$ . By considering its regularized problem, using Moser iteration technique, the local bounded properties of the $$L^{\infty }$$ -norm of $$u_{k}$$ and that of the $$L^{p}$$ -norm of the gradient $$\nabla u_{k}$$ are got, where $$u_{k}$$ is the solution of the regularized problem of the equation. By the compactness theorem, the existence of the solution of the equation itself is obtained. By using some techniques in Zhao and Yuan (Chin Ann Math A 16(2):179–194, 1995), the stability of the solutions is obtained too.