Backward uniqueness for parabolic operators whose coefficients are non-Lipschitz continuous in time. (English) Zbl 1074.35042

The authors consider the backward parabolic operator \(L := \partial_t + \sum_{j,k=1}^n \partial_{x_j}(a_{jk}(t,x)\partial_{x_k}) + \sum_{j=1}^n b_j(t,x) \partial _{x_j} + c(t,x)\). All the coefficients are defined in \([0,T]\times \mathbb{R}_x^n\), measurable and bounded; the coefficients \(b_j\) and \(c\) are complex valued; \((a_{jk}(t,x))_{jk}\) is a real matrix for all \((t,x) \in [0,T]\times \mathbb{R}_x^n\) and there is \(\lambda_0 \in (0,1]\) such that \(\sum_{j,k=1}^n a_{jk}(t,x) \xi_j\xi_k \geq \lambda_0 | \xi| ^2\) for all \((t,x)\in [0,T]\times \mathbb{R}_x^n\) and \(\xi \in \mathbb{R}_\xi^n\). Given a functional space \({\mathcal H}\) (in which it makes sense to look for the solutions of the equation \(Lu =0\)) one says that the operator \(L\) has the \({\mathcal H}\)-uniqueness property if, whenever \(u \in {\mathcal H}, Lu =0\) in \([0,T]\times \mathbb{R}_x^n\) and \(u(0,x) = 0\) in \(\mathbb{R}_x^n\), then \(u=0\) in \([0,T]\times \mathbb{R}_x^n\). With \({\mathcal H}_1 := H^1([0,T],L^2(\mathbb{R}^n_x))\cap L^2([0,T],H^2(\mathbb{R}^n_x))\) the authors prove the \({\mathcal H}_1\)-uniqueness property for \(L\) when the coefficients \(a_{jk}\) are \(C^2\) in the \(x\) variables and non-Lipschitz-continuous in \(t\). The regularity in \(t\) is given in terms of a modulus of continuity \(\eta\) satisfying the Osgood condition \(\int_0^1 \frac{ds}{\eta(s)} = +\infty\). It is proved that this condition is also necessary.


35K15 Initial value problems for second-order parabolic equations
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