Communications on Pure & Applied Analysis最新文献

This paper deals with an exponentially decaying diffusive chemotaxis system with indirect signal production or consumption

begin{document}$ begin{eqnarray*} label{1a} left{ begin{split}{} &u_t = nablacdot(D(u)nabla u)-nablacdot(S(u)nabla v), &(x,t)in Omegatimes (0,infty), &v_t = Delta v+h(v,w), &(x,t)in Omegatimes (0,infty), &w_t = Delta w- w+u, &(x,t)in Omegatimes (0,infty), end{split} right. end{eqnarray*} $end{document}

under homogeneous Neumann boundary conditions in a smoothly bounded domain begin{document}$ Omegasubset mathbb{R}^{n} $end{document}, begin{document}$ ngeq2 $end{document}, where the nonlinear diffusivity begin{document}$ D $end{document} and chemosensitivity begin{document}$ S $end{document} are supposed to satisfy

begin{document}$ K_{1}e^{-beta^{-}s}leq D(s) leq K_{2}e^{-beta^{+}s} ;;;{rm{and}};;;frac{D(s)}{S(s)}geq K_{3}s^{-alpha}+gamma, $end{document}

with the constants begin{document}$ beta^{-}geq beta^{+}>0 $end{document}, begin{document}$ K_{1},K_{2},K_{3}>0 $end{document} and begin{document}$ alpha,gammageq0 $end{document}. When begin{document}$ h(v,w) = -v+w $end{document}, we study the global existence and boundedness of solutions for the above system provided that begin{document}$ alphain[0,frac{2}{n}) $end{document}, begin{document}$ beta^{-}geq beta^{+}>frac{n}{2} $end{document}, begin{document}$ gamma>1 $end{document} and the initial mass of begin{document}$ u_{0} $end{document} is small enough. Moreover, it is proved that the global bounded solution begin{document}$ (u,v,w) $end{document} converges to begin{document}$ (overline{u_{0}},overline{u_{0}},overline{u_{0}}) $end{document} in the begin{document}$ L^{infty} $end{document}-norm as begin{document}$ trightarrow

This paper is devoted to investigating formation of singularities for solutions to semilinear Moore-Gibson-Thompson equations with power type nonlinearity begin{document}$ |u|^{p} $end{document}, derivative type nonlinearity begin{document}$ |u_{t}|^{p} $end{document} and combined type nonlinearities begin{document}$ |u_{t}|^{p}+|u|^{q} $end{document} in the case of single equation, combined type nonlinearities begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $end{document}, begin{document}$ |u_{t}|^{p_{2}}+|u|^{q_{2}} $end{document}, combined and power type nonlinearities begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $end{document}, begin{document}$ |u|^{q_{2}} $end{document}, combined and derivative type nonlinearities begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $end{document}, begin{document}$ |u_{t}|^{p_{2}} $end{document} in the case of coupled system, respectively. More precisely, blow-up results of solutions to problems in the sub-critical and critical cases are derived by applying test function technique. Moreover, upper bound lifespan estimates of solutions to the coupled systems are investigated. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent.

This paper is concerned with second-order elliptic operators whose diffusion coefficients degenerate at the boundary in first order. In this borderline case, the behavior strongly depends on the size and direction of the drift term. Mildly inward (or outward) pointing and strongly outward pointing drift terms were studied before. Here we treat the intermediate case equipped with Dirichlet boundary conditions, and show generation of an analytic positive begin{document}$ C_0 $end{document}-semigroup. The main result is a precise description of the domain of the generator, which is more involved than in the other cases and exhibits reduced regularity compared to them.

In this paper, we deal with the backward problem for nonlinear parabolic equations involving a pseudo-differential operator in the begin{document}$ n $end{document}-dimensional space. We prove that the problem is ill-posed in the sense of Hadamard, i.e., the solution, if it exists, does not depend continuously on the data. To regularize the problem, we propose two modified versions of the so-called optimal filtering method of Seidman [T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162–170]. According to different a priori assumptions on the regularity of the exact solution, we obtain some sharp optimal estimates of the Hölder-Logarithmic type in the Sobolev space begin{document}$ H^q(mathbb{R}^n) $end{document}.

In this paper, we consider the following general pseudo-relativistic Schrödinger equation with indefinite nonlinearities:

begin{document}$ (-Delta+m^{2})^{s}u = a(x_1)fleft(u,nabla uright),quad {rm{in}} ,,mathbb R^{N}, $end{document}

where begin{document}$ sin(0,1) $end{document}, mass begin{document}$ m>0 $end{document} and begin{document}$ a $end{document} is a non-decreasing functions. We prove the nonexistence and the monotonicity of the positive bounded solution for the above equation via the direct method of moving planes.

This paper is concerned with the global solutions of the 3D compressible micropolar fluid model in the domain to a subset of begin{document}$ R^3 $end{document} bounded with two coaxial cylinders that present the solid thermo-insulated walls, which is in a thermodynamical sense perfect and polytropic. Compared with the classical Navier-Stokes equations, the angular velocity begin{document}$ w $end{document} in this model brings benefit that is the damping term -begin{document}$ uw $end{document} can provide extra regularity of begin{document}$ w $end{document}. At the same time, the term begin{document}$ uw^2 $end{document} is bad, it increases the nonlinearity of our system. Moreover, the regularity and exponential stability in begin{document}$ H^4 $end{document} also are proved.