## Differential and Integral Equations

- Differential Integral Equations
- Volume 11, Number 1 (1998), 179-190.

### Global existence and nonexistence for a parabolic system with nonlinear boundary conditions

Julio D. Rossi and Noemi Wolanski

#### Abstract

We find both necessary and sufficient conditions on the nonnegative increasing functions $f$ and $g$ so that positive solutions of $$ \begin{matrix} u_t=\Delta u+h(u,v)\qquad &v_t=\Delta v+r(u,v) \qquad &\text {in }\Omega\times (0,T)\\ \frac{\partial}{\partial\nu} u=f(v) \qquad &\frac{\partial}{\partial\nu} v=g(u)\qquad &\text {on } \partial\Omega\times (0,T) \end{matrix} $$ exist globally in time. We assume throughout that $h$ and $r$ are nonnegative, smooth and $\frac{h(u,v)}u$, $\frac{r(u,v)}v$ are globally bounded.

#### Article information

**Source**

Differential Integral Equations, Volume 11, Number 1 (1998), 179-190.

**Dates**

First available in Project Euclid: 1 May 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1367414142

**Mathematical Reviews number (MathSciNet)**

MR1608013

**Zentralblatt MATH identifier**

1004.35012

**Subjects**

Primary: 35K60: Nonlinear initial value problems for linear parabolic equations

Secondary: 35B60: Continuation and prolongation of solutions [See also 58A15, 58A17, 58Hxx]

#### Citation

Rossi, Julio D.; Wolanski, Noemi. Global existence and nonexistence for a parabolic system with nonlinear boundary conditions. Differential Integral Equations 11 (1998), no. 1, 179--190. https://projecteuclid.org/euclid.die/1367414142