TAILIEUCHUNG - Blow-up phenomena for nonlocal inhomogeneous diffusion problems
This paper is concerned with the blow-up of solutions to some nonlocal inhomogeneous dispersal equations subject to homogeneous Neumann boundary conditions. We establish conditions on nonlinearities sufficient to guarantee that solutions exist for all time as well as blow up at some finite time. Moreover, lower bounds for blow-up time of nonlocal problems are obtained. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 466 – 482 ¨ ITAK ˙ c TUB doi: Blow-up phenomena for nonlocal inhomogeneous diffusion problems Jian-Wen SUN,∗ Fei-Ying YANG School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, P. R. China Received: • Accepted: • Published Online: • Printed: Abstract: This paper is concerned with the blow-up of solutions to some nonlocal inhomogeneous dispersal equations subject to homogeneous Neumann boundary conditions. We establish conditions on nonlinearities sufficient to guarantee that solutions exist for all time as well as blow up at some finite time. Moreover, lower bounds for blow-up time of nonlocal problems are obtained. Key words: Blow-up solutions, bounds on blow-up time, nonlocal dispersal, comparison principle 1. Introduction and main results During the past twenty years, the nonlocal diffusion equation of the form J(x, y)u(y, t)dy − u(x, t), ut (x, t) = RN and variations of it, have been widely used to model diffusion process. The research on the nonlocal diffusion equations has attracted some attention; see [16, 3, 2, 8, 23, 6, 5, 4, 15, 25]. For the study of evolution equations, one of the most remarkable properties is to consider the blow-up problems; that is, the solutions may become unbounded in finite time, and such phenomena are known as blow-up in the literature. If our model describes a physical or chemical reaction process which may become discontinuous before blow-up, then the bounds for blow-up time are very useful and a lower bound gives a safe time interval for operation or reaction. A variety of methods have been used to study the questions like existence and nonexistence of global solutions, blowup solutions, upper estimates of blow-up solutions, blow-up times, blow-up rates and asymptotic behaviors of classical parabolic problems (see [26, 22, .
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