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Log-log convexity and backward uniqueness. (English) Zbl 1119.35007

The author investigates the so called backward uniqueness problem for evolution equations. That is, one considers an evolution equation
\[ \partial_t u+Au=f(u)\tag{1} \]
(on a Banach space \(X\)) subject to some restrictions; one is given a solution \(u(t):t\in [0,T]\) such that \(u(T)=0\). One now seeks conditions on \(A\), \(f\), \(u\) which guarantee that \(u(t)=0\) for \(t\in [0,T]\) holds. This question has been treated by various authors. Here the author investigates this problem under the following assumptions. On a real or complex Hilbert space \(H\) with scalar product \((\,,\,)\) and norm \(\|\;\|\) one has the evolution equation \[ \partial_t u+Au=f,\quad f\in C([T_0,0],H)\tag{2} \] subject to the following stipulations: \(A\) is symmetric, with \({\mathcal D}(A)\subseteq H\) and such that \((Av,v)\geq 0\), \(v\in{\mathcal D}(A)\). Moreover, one has a solution \(u(\;)\) of (2), ie.: \[ u\in C^1([T_0,0],H);\quad u(t)\in{\mathcal D}(A)\text{ for }t\in [T_0,0]\text{ and }Au(\;)\in C([T_0,0],H),\tag{3} \] and \(u(\;)\) satisfies (2) pointwise. Next let \[ M_0\geq 2\sup_{t\in[T_0,0]}\|u(t)\| \] and set \(L(x)=\log(M^2_0/x^2)\) and \(\|A^{1/2}w\|=(Aw,w)^{1/2}\), \(w\in{\mathcal D}(A)\). The function \(f(t)\), \(t\in [T_0,0]\) is assumed to satisfy two inequalities, the first of which is as follows: \[ \|f\|\leq\frac{K_1}{L(\|u\|)^{\beta/2}}\;\|A^{1/2}u\|^{1-\beta}\|Au\|^\beta+K_2 L(\|u\|)^{\alpha/2}\|u\|,\quad u=u(t)\tag{4} \]
on \(t\in [T_0,0]\), some \(\alpha,\beta\in [0,1]\), \(K_1,K_2\geq 0\) and with \(K^2_1\leq \alpha/8\) if \(\beta=1\). The second inequality is similar but more involved. The main result is Theorem 2.1 which states that under the above conditions \(u(0)=0\) implies \(u(t)=0\) for \(t\in[T_0,0]\). The proof proceeds by a rather intricate series of identities and inequalities which cannot be discussed here. A central role is thereby played by the expression
\[ \widetilde Q(t)=Q(t)L(\|u(t)\|)^{-\alpha}, \]
where \(Q(t)=\|A^{1/2}u(t)\|^2 \|u(t)\|^{-2}\) is the Dirichlet quotient which appears in earlier investigations of the backward problem. The paper concludes with three applications to PDE’s, two of which are to parabolic PDE’s, while the third is to the \(2d\)-Navier-Stokes equation in \(2\pi\)-periodic setting. That is one considers the system
\[ \partial_t u-\Delta u+(u\nabla)u+\nabla p=f\text{ on }\Omega=[0,2\pi]^2,\;\text{div\,}u=0\tag{5} \]
with \(2\pi\)-periodic boundary conditions and with \(f\) time independent. It is known that (5) admits a global attractor \(\mathcal A\). Theorem 3.1 then states that the following relation holds: \[ \sup_{u_1,u_2}\frac{\|\nabla(u_1-u_2)\|^2}{\|u_1-u_2\|^2\,{\mathcal D}(u_1,u_2)}<\infty\text{ where }\|\;\|=\|\;\|_{{\mathcal L}^2_{\text{per}}},\;u_1,u_2\in{\mathcal A},\;u_1\neq u_2\tag{6} \]
while \({\mathcal D}(u_1,u_2)=\log(M^2_0/\|u_1-u_2\|^2)\) and \(M_0=4\sup\|u\|\), \(u\in{\mathcal A}\). Whether (6) holds without the factor \({\mathcal D}(u_1,u_2)\) is an open problem.

MSC:

35B42 Inertial manifolds
35B41 Attractors
35K55 Nonlinear parabolic equations
34G20 Nonlinear differential equations in abstract spaces
35K15 Initial value problems for second-order parabolic equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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