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 $\sum\limits_{j,k=1}^n a_{jk}(t,x)\xi_j\xi_k\geq \kappa \xi ^2, \quad \forall \, (t,x,\xi) \in [0,T] \times {\mathbb R}^n_x \times {\mathbb R}^n_\xi.$ 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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