In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem [ u_{t}=Delta u+displaystylefrac{lambda f(u)}{big(int_{Omega}f(u)dxbig)^{p}}, xin Omega, t>0, ] with homogeneous Dirichlet boundary condition, where $lambda>0, p>0$,
$f$ is nonincreasing. It is found that: (a) For $0<pleq1$, $u(x,t)$ is globally bounded and the unique stationary solution is globally asymptotically stable for any $lambda>0$; (b) For $1<p<2$, $u(x,t)$ is globally bounded for any $lambda>0$; (c) For $p=2$, if $0<lambda<2|partialOmega|^2$, then $u(x,t)$ is globally bounded, if $lambda=2|partialOmega|^2$, there is no stationary solution and $u(x,t)$ is a global solution and $u(x,t)toinfty$ as $ttoinfty$ for all $xinOmega$, if $lambda>2|partialOmega|^2$, there is no stationary solution and $u(x,t)$ blows up in finite time for all $xinOmega$; (d) For $p>2$, there exists a $lambda^*>0$ such that for $lambda>lambda^*$, or for $0<lambdaleqlambda^*$ and $u_0(x)$ sufficiently large, $u(x,t)$ blows up in finite time. Moreover, some formal asymptotic estimates for the behavior of $u(x,t)$ as it blows up are obtained for $pgeq2$.
In this paper, we present a bootstrap procedure for general elliptic systems with $n(geq 3)$ components. Combining with the $L^p$-$L^q$-estimates, it yields the optimal $L^infty$-regularity conditions for the three well-known types of weak solutions:
$H_0^1$-solutions, $L^1$-solutions and $L^1_delta$-solutions. Thanks to the linear theory in $L^p_delta(Omega)$, it also yields the optimal conditions for a priori estimates for $L^1_delta$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.
In this paper we present a new bootstrap procedure for elliptic systems with two unknown functions. Combining with the $L^p$-$L^q$-estimates, it yields the optimal $L^infty$-regularity conditions for the three well-known types of weak solutions: $H_0
^1$-solutions, $L^1$-solutions and $L^1_delta$-solutions. Thanks to the linear theory in $L^p_delta(Omega)$, it also yields the optimal conditions for a priori estimates for $L^1_delta$-solutions. Based on the a priori estimates, we improve known existence theorems for some classes of elliptic systems.
