Conditional stability for backward parabolic equations with LogLipt x Lipx-coefficients
speaker: Daniele Del Santo (Università di Trieste, Dipartimento di Matematica e Geoscienze)abstract:
Consider the equation
\[
Pu=\partial_t u
+\sum_{i,j}\partial_{x_i}(a_{i,j}(t,x)\partial_{x_j}u)+\sum_j
b_j(t,x)u\partial_{x_j}+c(t,x)u=0
\]
on the strip \([0,T] \times {\mathbb R}^n_x\).
The coefficients are supposed to be real valued, measurable and bounded. The matrix \((a_{jk})_{j,k=1,\dots, n}\) is symmetric and positive definite, i.e.
there exists a \(\kappa>0\) such that
\[
\sum\limits_{j,k=1}^n a_{jk}(t,x)\xi_j\xi_k\geq \kappa
\xi
^2, \quad \forall \, (t,x,\xi) \in [0,T] \times {\mathbb R}^n_x \times {\mathbb R}^n_\xi.
\]
It is well known that the Cauchy problem for the equation above, when
the data are given on \(\{t=0\}\), is an ill-posed problem. Nevertheless a result due to Hurd says that:
for every \(T<0\) and for every \(T'\in\,\,]0,T[\) and \(D>0\) there exist \(\rho>0\),
\(0<\delta<1\) and \(M>0\) such that, if \(u\in C^0([0,T], L^2({\mathbb R}^n))\cap C^0([0,T[,
H^1({\mathbb R}^n))\cap C^1([0,T[, L^2({\mathbb R}^n))\) is a
solution of \(Pu\equiv0\) on \([0,T]\) with
\(\
u(0,\cdot)\
_{L^2}\leq\rho\) and \(\
u(t,\cdot)\
_{L^2}\leq D\) on
\([0,T]\), then
\[\sup_{t\in[0,T']}\
u(t,\cdot)\
_{L^2}\leq
M\
u(0,\cdot)\
_{L^2}^\delta.
\]
The constants \(\rho\), \(M\) and \(\delta\) depend only on \(T\), \(T'\) and \(D\), on the ellipticity constant of \(P\), on the \(L^\infty\) norms of the coefficients \(a_{i,j}\)'s, \(b_i\)'s, \(c\) and of their spatial derivatives, and on the Lipschitz constant of the coefficients \(a_{i,j}\)'s with respect to time.
In this seminar I will consider the situation under the hypothesis that the coefficients are non Lipschitz-continuous.
timetable:
Wed 5 Oct, 9:30 - 10:15, Sala Stemmi
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