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28 March 2024
 
  » arxiv » 1703.0834

 Article overview


Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term
Martina Magliocca ;
Date 2 Mar 2017
AbstractWe study the existence of solutions of a nonlinear parabolic problem of Cauchy-Dirichlet type having a lower order term which depends on the gradient. The model we have in mind is the following: [ egin{cases}egin{split} & u_t- ext{div}(A(t,x) abla u| abla u|^{p-2})=gamma | abla u|^q+f(t,x) &qquad ext{in } Q_T,\ & u=0 &qquad ext{on }(0,T) imes partial Omega,\ & u(0,x)=u_0(x) &qquad ext{in } Omega, end{split}end{cases} ] where $Q_T=(0,T) imes Omega$, $Omega$ is a bounded domain of $mathrm{R}^N$, $Nge 2$, $1<p<N$, the matrix $A(t,x)$ is coercive and with measurable bounded coefficients, the r.h.s. growth rate satisfies the superlinearity condition [ maxleft{frac{p}{2},frac{p(N+1)-N}{N+2} ight}<q<p ] and the initial datum $u_0$ is an unbounded function belonging to a suitable Lebesgue space $L^sigma(Omega)$. We point out that, once we have fixed $q$, there exists a link between this growth rate and exponent $sigma=sigma(q,N,p)$ which allows one to have (or not) an existence result. Moreover, the value of $q$ deeply influences the notion of solution we can ask for. The sublinear growth case with [ 0<qlefrac{p}{2} ] is dealt at the end of the paper for what concerns small value of $p$, namely $1<p<2$.
Source arXiv, 1703.0834
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