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Article overview
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Existence results for a Cauchy-Dirichlet parabolic problem with a repulsive gradient term | Martina Magliocca
; | Date: |
2 Mar 2017 | Abstract: | We 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 | Services: | Forum | Review | PDF | Favorites |
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