Abstract
We prove the uniqueness for backward parabolic equations whose coefficients are Osgood continuous in time for t>0 but not at t=0.
Highlights
Let us consider the following backward parabolic operator ∑n L= t+xj aj,k(t, x) xk + bj(t, x) xj + c(t, x), (1)j,k=1 j=1 where all the coefficients are assumed to be defined in [0, T] × Rn, measurable and bounded; (aj,k(t, x))j,k is a real symmetric matrix for all (t, x) ∈ [0, T] × Rn and there exists 0 ∈ (0, 1] such that∑n aj,k(t, x) j k ≥ 0| |2, (2)j,k=1 for all (t, x) ∈ [0, T] × Rn and ∈ Rn
In [9, 10], we investigated the problem of finding the minimal regularity assumptions on the coefficients aj,k ensuring the H–uniqueness property to (1)
We show that if the loss of the Osgood continuity is properly controlled as t goes to 0, the H–uniqueness property for (1) remains valid
Summary
J,k=1 j=1 where all the coefficients are assumed to be defined in [0, T] × Rn , measurable and bounded; (aj,k(t, x))j,k is a real symmetric matrix for all (t, x) ∈ [0, T] × Rn and there exists 0 ∈ A counterexample in [9], similar to that one of Pliś quoted here above, shows that, considering the regularity with respect to t for the aj,k , the Osgood condition is sharp: given any non-Osgood modulus of continuity , it is possible to construct a backward parabolic operator like (1), whose coefficients are C∞ in x and -continuous in t, for which the H –uniqueness property does not hold. The coefficients aj,k are assumed to be globally Lipschitz continuous in x Under such hypothesis, we prove that the H–uniqueness property holds for (1). The borderline case = 0 in (5) is considered in paper [11] In such a situation, only a very particular uniqueness result holds and the problem remains essentially open
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