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Global existence and blow-up problems for reaction diffusion model with multiple nonlinearities. (English) Zbl 1140.35010
Summary: We study the following reaction diffusion model, $$u_t= \nabla\cdot(a(u) b(x) c(t)\nabla u)+ g(x,t) f(u)\qquad\text{in }D\times (0,T),$$ $${\partial u\over\partial n}= h(x, t)r(u)\qquad\text{on }\partial D\times (0,T),$$ $$u(x,0)= u_0(x)> 0\qquad\text{in }\overline D,$$ where $D$ is a bounded domain $\bbfR^N$ with smooth boundary $\partial D$, $N\ge 2$. This paper deals with interactions among three kinds of nonlinear mechanisms: nonlinear reaction, nonlinear convection and nonlinear boundary flux. The existence theorems of blow-up positive solutions, upper bound of “blow-up time”, upper estimates of “blow-up rate”, existence theorems of global positive solutions, and upper estimates of global positive solutions are given under suitable assumptions on $a$, $b$, $c$, $f$, $g$, $h$, $r$ and initial data $u_0(x)$.

##### MSC:
 35K60 Nonlinear initial value problems for linear parabolic equations 35K57 Reaction-diffusion equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
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##### References:
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