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.

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 L², and the initial density, the gradients of initial velocity/liquid crystal director field are bounded in L^∞, L² and H¹, 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.

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 theL¹-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.

We prove the convergence of the solutions $$u_{m,p}$$ of the equation $$u_t+(u^m)_x=-\,u^p$$ in $${{\mathbb {R}}}\times (0,\infty )$$ , $$u(x,0)=u_0(x)\ge 0$$ in $${{\mathbb {R}}}$$ , as $$m\rightarrow \infty $$ for any $$p>1$$ and $$u_0\in L^1({{\mathbb {R}}})\cap L^{\infty }({{\mathbb {R}}})$$ or as $$p\rightarrow \infty $$ for any $$m>1$$ and $$u_0\in L^{\infty }({{\mathbb {R}}})$$ . We also show that in general $$\underset{p\rightarrow \infty }{\lim }\underset{m\rightarrow \infty }{\lim }u_{m,p}\ne \underset{m\rightarrow \infty }{\lim }\underset{p\rightarrow \infty }{\lim }u_{m,p}$$ .

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 u₀ and u₁ satisfy suitable assumptions.

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

The aims and objectives of this manuscript are concerned with the investigation of some appropriate conditions to establish existence theory of solutions to a class of nonlinear four-point boundary value problem (BVP) corresponding to fractional order differential equations (FODEs) provided as $$\begin{aligned} \left\{ \begin{aligned} ^{c}&{\mathscr {D}}^{\omega }y(t)={\mathcal {F}}\left(t, y(t), {^{c}{\mathscr {D}}^{\omega -1}y(t)}\right),1<\omega \le 2,\,\, t\in {\mathbf{J }}=[0, 1],\\&y(0)=\zeta y(\alpha ), \,\, y(1)=\xi y(\beta ),\,\xi ,\ \zeta ,\ \alpha ,\,\beta \in (0,1), \end{aligned}\right. \end{aligned}$$ where $${^{c}{\mathscr {D}}^{\omega }}$$ is Caputo’s fractional derivative of order q and $${\mathcal {F}}\in ({\mathbf {J}}\times {\mathbf {R}} \times {\mathbf {R}} ,{\mathbf {R}})$$ may be nonlinear. The required conditions are obtained by using classical results of functional analysis and fixed point theory. Further, we establish some adequate conditions for the Ulam–Hyers stability and generalized Ulam–Hyers stability for the solutions to the considered BVP of nonlinear FODEs. We include a proper problem to illustrate our established results.

In this paper, we study the following Choquard equations with small perturbation f$$-\Delta u + V(x)u = (I_\alpha * |u|^p)|u|^{p-2}u+f(x), x\in \mathbb{R}^N.$$ where N ≥ 3 and I_α denotes the Riesz potential. As is known that the above equation has a ground state u_α and a bound state v_α by fibering maps (see [22] or [23]), our aim is to show that for fixed $$p \in (1,\frac{N}{N-2})$$, u_α and v_α converge to a ground state and a bound state of the limiting local problem respectively, as α → 0.

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.

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 $$f$$ .

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 $$ H^{3}_{0} {\left( {0,1} \right)} \times H^{1}_{0} {\left( {0,1} \right)} $$ .