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This paper is concerned with traveling curved fronts in reaction diffusion equations with degenerate monostable and combustion nonlinearities. For a given admissible pyramidal in three-dimensional spaces, the existence of a pyramidal traveling front has been proved by Wang and Bu [30] recently. By constructing new supersolutions and developing the arguments of Taniguchi [25] for the Allen-Cahn equation, in this paper we first characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces, and then establish the uniqueness and asymptotic stability of such three-dimensional pyramidal traveling fronts under the condition that given perturbations decay at infinity.

In this paper we consider the global stability of traveling curved fronts in the Belousov–Zhabotinskii (BZ for short) system in R 2 . The existence and part of the stability of such curved fronts have already been proved in our recent work (Niu et al., 2018). The aim of this paper is to prove the global asymptotic stability of traveling curved fronts in space. The degeneration of the system at one of its equilibria makes its stability hard to study. We overcome this difficulty by introducing a mild subsolution which involves a perturbation that is the power of the one-dimensional traveling front.

In this paper, we study the pulsating fronts of reaction–advection-diffusion equations with two types of nonlinear term in periodic excitable media. Firstly, for the case with combustion nonlinearity, the unique front is proved to decay exponentially when it approaches the unstable limiting state. Secondly, for the degenerate monostable type nonlinearity, it is shown that the front with critical speed is unique, monotone and decays exponentially at negative end, while the fronts of noncritical speeds decay to zero non-exponentially.

This paper is concerned with combustion transition fronts in $$\mathbb {R}^{N}$$$$(N\ge 1)$$. Firstly, we prove the existence and the uniqueness of the global mean speed which is independent of the shape of the level sets of the fronts. Secondly, we show that the planar fronts can be characterized in the more general class of almost-planar fronts. Thirdly, we show the existence of new types of transitions fronts in $$\mathbb {R}^{N}$$ which are not standard traveling fronts. Finally, we prove that all transition fronts are monotone increasing in time, whatever shape their level sets may have.

In this paper we consider a diffusion system with the Belousov–Zhabotinskii (BZ for short) chemical reaction. Following Brazhnik and Tyson [4] and Perez-Munuzuri et al. [45] , who predicted V-shaped fronts theoretically and discovered V-shaped fronts by experiments respectively, we give a rigorous mathematical proof of their results. We establish the existence of V-shaped traveling fronts in R 2 by constructing a proper supersolution and a subsolution. Furthermore, we establish the stability of the V-shaped front in R 2 .

This paper is concerned with the global stability of V-shaped traveling fronts in reaction-diffusion equations with combustion and degenerate monostable nonlinearity. The existence of such curved fronts has been recently proved by [ 39 ]. In this paper, by constructing new subsolutions, we show the asymptotic stability of V-shaped traveling fronts.

For convex body M and star body Q in R n , we introduce a new concept of the ( p , q ) -projection body Π p , q ( M , Q ) , as an extension of L p -projection body Π p M . For p ≥ 1 and convex bodies K , L in R n , the L p -Shephard problem asks whether Π p K ⊆ Π p L implies that | K | n − p ≤ | L | n − p ? Here | M | is the volume of M. It has shown in [21] , [22] , [24] that the answer is affirmative when n ≤ 2 and negative when n ≥ 3 . In this paper, we consider a variant of the L p -Shephard problem via ( p , q ) -projection bodies, i.e., if Π p K ⊆ Π p , q ( M , Q ) ⊆ Π p L , for any fixed Q and q ∈ R , can we compare the volume of K , M and L? The problem includes two parts with comparing | M | and | L | when Π p , q ( M , Q ) ⊆ Π p L , and comparing | K | and | M | when Π p K ⊆ Π p , q ( M , Q ) . This paper answers two parts of the question by connecting with L p -projection body, and Levy's embedding L p . Note that in the case of q = n (or Q = M ) the question is exactly the classic L p -Shephard problem.

This paper is concerned with traveling curved fronts in reaction diffusion equations with degenerate monostable and combustion nonlinearities. For a given admissible pyramidal in three-dimensional spaces, the existence of a pyramidal traveling front has been proved by Wang and Bu [ 30 ] recently. By constructing new supersolutions and developing the arguments of Taniguchi [ 25 ] for the Allen-Cahn equation, in this paper we first characterize the pyramidal traveling front as a combination of planar fronts on the lateral surfaces, and then establish the uniqueness and asymptotic stability of such three-dimensional pyramidal traveling fronts under the condition that given perturbations decay at infinity.

In this paper, multidimensional stability of pyramidal traveling fronts are studied to the reaction-diffusion equations with degenerate Fisher-KPP monostable and combustion nonlinearities. By constructing supersolutions and subsolutions coupled with the comparison principle, we firstly prove that under any initial perturbation (possibly large) decaying at space infinity, the three-dimensional pyramidal traveling fronts are asymptotically stable in weighted \begin{document}$ L^{\infty} $\end{document} spaces on \begin{document}$ \mathbb{R}^{n}\; (n\geq4) $\end{document}. Secondly, we show that under general bounded perturbations (even very small), the pyramidal traveling fronts are not asymptotically stable by constructing a solution which oscillates permanently between two three-dimensional pyramidal traveling fronts on \begin{document}$ \mathbb{R}^{4} $\end{document}.

This paper is concerned with the reaction–advection–diffusion equations with bistable nonlinearity in periodic media. Assume that the equation has three equilibria: an unstable equilibrium θ and two stable equilibria 0 and 1. It is known that there exist different pulsating fronts connecting any two of those three equilibria. In this paper we first study the exponential behavior of the fronts when they approach their stable limiting states. Then, we establish three different types of pulsating entire solutions for the equation. To establish the existence of entire solutions, we consider combinations of any two of those different pulsating fronts and construct appropriate sub- and supersolutions.

This paper is concerned with nonplanar traveling fronts in reaction–diffusion equations with combustion nonlinearity and degenerate Fisher-KPP nonlinearity. Our study contains two parts: in the first part we establish the existence of traveling fronts of pyramidal shape in R 3 , and in the second part we establish the existence and stability of V-shaped traveling fronts in R 2 .

This paper is concerned with conical traveling fronts for reaction–diffusion equations with combustion nonlinearity in R 3 . By constructing a sequence of pyramidal traveling fronts and taking its limit, we establish the existence and qualitative properties of conical traveling fronts.

This paper is concerned with the multidimensional stability of traveling fronts for the combustion and non-KPP monostable equations. Our study contains two parts: in the first part, we first show that the two-dimensional V-shaped traveling fronts are asymptotically stable in $$\mathbb {R}^{n+2}$$ with $$n\ge 1$$ under any (possibly large) initial perturbations that decay at space infinity, and then, we prove that there exists a solution that oscillates permanently between two V-shaped traveling fronts, which implies that even very small perturbations to the V-shaped traveling front can lead to permanent oscillation. In the second part, we establish the multidimensional stability of planar traveling front in $$\mathbb {R}^{n+1}$$ with $$n\ge 1$$ .

This paper is to study traveling fronts of reaction-diffusion equations with space-time periodic advection and nonlinearity in $\mathbb{R}^N$ with $N\geq3$. We are interested in curved fronts satisfying some ``pyramidal" conditions at infinity. In $\Bbb{R}^3$, we first show that there is a minimal speed $c^{*}$ such that curved fronts with speed $c$ exist if and only if $c\geq c^{*}$, and then we prove that such curved fronts are decreasing in the direction of propagation. Furthermore, we give a generalization of our results in $\mathbb{R}^N$ with $N\geq4$.