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August  2017, 10(4): 837-852. doi: 10.3934/dcdss.2017042

## On the existence of weak solutions of an unsteady p-Laplace thermistor system with strictly monotone electrical conductivities

 Department of Mathematics, Humboldt University Berlin, Unter den Linden 6,10099 Berlin, Germany

Received  April 2016 Revised  December 2016 Published  April 2017

Let
 $\Omega\subset\mathbb{R}^n$
(
 $n=2$
or
 $n=3$
) be a bounded domain. We consider the thermistor system
 $\text{(1)}\quad \nabla\cdot \boldsymbol{J}=0,\qquad \text{(2)}\quad \frac{\partial u}{\partial t}+\nabla\cdot\boldsymbol{q}=f(x,t,u,\nabla\varphi)\;\text{ in }\; \Omega\times\,]\,0,T\,[\,,$
where (1) is a
 $p$
-Laplace type equation for
 $\varphi$
(
 $u=$
temperature,
 $\varphi=$
electrostatic potential). We prove the existence of a weak solution
 $(\varphi,u)$
of (1)–(2) under mixed boundary conditions for
 $\varphi$
, and a Robin boundary condition and an initial condition for
 $u$
.
Citation: Joachim Naumann. On the existence of weak solutions of an unsteady p-Laplace thermistor system with strictly monotone electrical conductivities. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 837-852. doi: 10.3934/dcdss.2017042
##### References:

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