Non-simultaneous blow-up for a reaction-diffusion system with absorption and coupled boundary flux

This paper deals with non-simultaneous blow-up for a reaction-diffusion system with absorption and nonlinear boundary flux. We establish necessary and sufficient conditions for the occurrence of non-simultaneous blow-up with proper initial data

Coexistence of a diffusive predator–prey model with Holling type-II functional response and density dependent mortality

AbstractIn this paper, we consider a two competitor–one prey diffusive model in which both competitors exhibit Holling type-II functional response and one of the competitors exhibits density dependent mortality rate. First, we study the local and global existence of strong solution by using the C0 analytic semigroup. Then, we consider the local and global stability of the positive constant equilibrium by using the linearization method and Laypunov functional method, respectively. Furthermore, we derive some results for the existence and non-existence of non-constant stationary solutions when the diffusion rate of a certain species is small or large. The existence of non-constant stationary solutions implies the possibility of pattern formation in this system

The Blow-Up Rate for a Strongly Coupled System of Semilinear Heat Equations with Nonlinear Boundary Conditions

AbstractThe paper deals with the blow-up rate of positive solutions to the system ut=uxx+ul11vl12, vt=vxx+ul21vl22 with boundary conditions ux(1,t)=(up11vp12)(1,t) and vx(1,t)=(up21vp22)(1,t). Under some assumptions on the matrices L=(lij) and P=(pij) and on the initial data u0,v0, the solution (u,v) blows up at finite time T, and we prove that maxx∈[0,1]u(x,t) (resp. maxx∈[0,1]v(x,t)) goes to infinity as (T−t)α1/2 (resp. (T−t)α2/2), where αi<0 are the solutions of (P−Id)(α1,α2)t=(−1,−1)t

Blow-up analysis for a semilinear parabolic equation with nonlinear memory and nonlocal nonlinear boundary condition

In this paper, we consider a semilinear parabolic equation $$u_t=\Delta u+u^q\int_0^tu^p(x,s)ds,\quad x\in \Omega,\quad t>0$$ with nonlocal nonlinear boundary condition $u|_{\partial\Omega\times(0,+\infty)}=\int_\Omega\varphi(x,y) u^l(y,t)dy$ and nonnegative initial data, where $p$, $q\geq 0$ and $l>0$. The blow-up criteria and the blow-up rate are obtained

Output regulation of nonlinear singularly perturbed systems

AbstractIn this paper, the state feedback regulator problem of nonlinear singularly perturbed systems is discussed. It is shown that, under standard assumptions, this problem is solvable if and only if a certain nonlinear partial differential equation is solvable. Once this equation is solvable, a feedback law which solves the problem can easily be constructed. The developed control law is applied to a nonlinear chemical process

Existence of sign-changing solution with least energy for a class of Kirchhoff-type equation in R^N

We consider the existence of least energy sign-changing (nodal) solution of Kirchhoff-type elliptic problems with general nonlinearity. Using a truncated technique and constrained minimization on the nodal Nehari manifold, we obtain that the Kirchhoff-type elliptic problem possesses one least energy sign-changing solution by applying a Pohožaev type identity. Moreover, the energy of the sign-changing solution is strictly more than the ground state energy

Blowup Properties for a Semilinear Reaction-Diffusion System with Nonlinear Nonlocal Boundary Conditions

We investigate the blowup properties of the positive solutions for a semilinear reaction-diffusion system with nonlinear nonlocal boundary condition. We obtain some sufficient conditions for global existence and blowup by utilizing the method of subsolution and supersolution