Note on uniqueness of solutions of differential inequalities of parabolic type (Q2523628)

The author considers the question of uniqueness of solutions of a parabolic boundary-value problem satisfying in addition certain differential inequalities. Specifically he considers the following boundary-value problem. Let \(l, m\) be positive integers such that \(l\) divides \(2m\) and set \(d = 2m/l\). Consider the differential operators \[ A(x, t; D_x, D_t) = D_t^l + \sum_{\substack{(\alpha,k) \\ \vert\alpha + kd\vert\le 2m \\ 0\le k< l}} a_{(\alpha,k)}(x, t) D_x^\alpha D_t^k \tag{1} \] \[ B_j(x, t; D_x, D_t) = D_t^l + \sum_{\substack{(\beta,k) \\ \vert\beta + kd\vert\le m_j \\ 0\le k< l_j}} b^j_{(\beta,k)}(x, t) D_x^\beta D_t^k \tag{2} \] where \(j = 1, \ldots, m\), \(m_j < 2m\), \(l_j = [m_j/d] = \) integral part of \(m_j/d\). Here \(x\in\Omega\) a bounded domain in \(\mathbb R^n\) and \(t\in (-\infty, T]\). Concerning the above operators the author makes the assumptions: I. \(A(x, t; D_x, D_t)\) is \(d\)-parabolic in \(\overline{\Omega}\times (-\infty, T]\), i. e., for fixed \(t\in (-\infty, T]\) and \(\vartheta\) with \(-\pi \le \vartheta \le 0\), \(A(x, t; D_x, e^{i\vartheta} D_y^d)\) is an elliptic operator for \((x, y)\in (\overline{\Omega}\times (-\infty, \infty)\). II. The ''Complementing Condition'' of \textit{S. Agmon}, \textit{A. Douglis} and \textit{L. Nirenberg} [Commun. Pure Appl. Math. 12, 623--727 (1959; Zbl 0093.10401)] for an elliptic boundary-value problem is satisfied by \(A(x, t; D_x, e^{i\vartheta} D_y^d)\) and \(B_j(x, t; D_x, e^{i\vartheta} D_y^d)\) for \((x, y)\in (\overline{\Omega}\times (-\infty, \infty)\), uniformly for \(t\in (-\infty, T]\) and \(-\pi \le \vartheta \le 0\). III. The coefficients of \(A(x, t; D_x, D_t)\) are uniformly continuous in \(\overline{\Omega}\times (-\infty,T]\) and the functions \(D_x^\alpha b_{(\beta,k)}^j(x, t)\) and \(D_t^l b_{(\beta,k)}^j(x, t)\) are uniformly continuous in \(\overline{\Omega}\times (-\infty,T]\) for \(\vert\alpha\vert \le 2m - m\) and \(i = 0,\ldots, l+1\). IV. \(\partial\Omega\) class \(C^{2m}\). Let \(\Vert u(x, t)\Vert_0 = \left(\int_\Omega \vert u(x, t)\vert^2 \,dx\right)^{1/2}\). Theorem. Suppose \(D_x^\alpha D_t^k u(x, t)\in L^2(\Omega)\) for \(\vert\alpha\vert + kd \le 2m\) and that the map \(t \rightarrow D_x^\alpha D_t^k u(x, t)\) is continuous from \((-\infty,T] \rightarrow L^2(\Omega)\). There exists \(\eta > 0\) such that if \(u(x, t)\) satisfies the differential inequality \[ \Vert A(x, t; D_x, D_t) u(x, t)\Vert_0 \le \eta\, \sum_{\vert\alpha\vert + kd=2m} \Vert D_x^\alpha D_t^k u(x, t)\Vert_0 + C \sum_{\vert\alpha\vert + kd<2m} \Vert D_x^\alpha D_t^k u(x, t)\Vert_0 \] together with the boundary conditions \(B_j(x,t; D_x, D_t) u(x,t) = 0\) for \(x\in\partial\Omega\), \(0 < t \le T\) and \(D_t^k u(x, 0) = 0\) for \(k = 0, \ldots, l-1\), then \(u(x, t) = 0\) in \(\Omega\times (0, T]\).