Abstract
We consider the initial and boundary value problem for the quasi-linear parabolic system
$\begin{alignat*}{2} \frac{\partial u}{\partial t}-\mathrm{div} \sigma\big(x,t,u(x,t),Du(x,t)\big) &=f &\text{on }&\Omega\times(0,T),\\ u(x,t)&=0&\text{on }&\partial\Omega\times(0,T),\\ u(x,0)&=u_0(x)\quad&\text{on }&\Omega \end{alignat*}$
for a function u : Ω×[0,T)→ℝm with T>0. Here, f∈Lp′(0,T;Wp′(Ω;ℝm)) for some p∈(2n/2n+2),∞, and u0∈L2(Ω, ℝm). We prove existence of a weak solution under classical regularity, growth, and coercivity conditions for σ but with only very mild monotonicity assumptions.
Citation
Norbert Hungerbühler. "Quasi-linear parabolic systems in divergence form with weak monotonicity." Duke Math. J. 107 (3) 497 - 520, 15 April 2001. https://doi.org/10.1215/S0012-7094-01-10733-3
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