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New trends in Partial Differential Equations

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 \[ \sum\limits_{j,k=1}^n a_{jk}(t,x)\xi_j\xi_k\geq \kappa
^2, \quad \forall \, (t,x,\xi) \in [0,T] \times {\mathbb R}^n_x \times {\mathbb R}^n_\xi. \] 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 \(\
_{L^2}\leq\rho\) and \(\
_{L^2}\leq D\) on \([0,T]\), then \[\sup_{t\in[0,T']}\
_{L^2}\leq M\
_{L^2}^\delta. \]

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.

Wed 5 Oct, 9:30 - 10:15, Sala Stemmi
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