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.

MSC:

 35K15 Initial value problems for second-order parabolic equations
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References:

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