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A variational approach to evolution problems with variable domains. (English) Zbl 1016.35031

The existence of weak solutions and their regularity are studied for the following parabolic equations in noncylindrical domains \[ \begin{cases} v_t+{\mathcal A}(t, x)v(t,x)= f(t,x),\;&(t,x)\in\Omega= \bigcup_{t\in [0,T]}\{t\}\times{\mathcal O}(t),\\ v=0,\;&(t,x)\in\Sigma= \bigcup_{t\in [0,T]}\{t\}\times\partial{\mathcal O}(t),\\ v(x,0)= v_0(x),\;&x\in{\mathcal O}(0).\end{cases}\tag{P} \] Here \(\{{\mathcal O}(t)\}_{t\in[0, T]}\) is a family of bounded domains in \(\mathbb{R}^d\) of class \(C^{1,1}\) uniformly in \(t\in [0,T]\), and the variation of \({\mathcal O}(t)\) is Hölder continuous in the following sense: there exist \(\gamma\in(0,1)\) and \(C> 0\) such that \(\sup_{x\in{\mathcal O}(t)} d(x,{\mathcal O}(s))\leq C|t-s|^\gamma\) for all \(s< t\). Fix an open ball \(D\) such that \([0,T]\times D\supset\overline\Omega\) and assume: \[ {\mathcal A}(t,x)\phi(x)= -\sum^d_{i,j=1} \partial_j(a_{i,j}(t, x)\partial_i\phi(x))+ \sum^d_{i=1} b_i(t, x)\partial_i\phi(x)+ c(t,x)\phi(x), \]
\[ a_{i,j}\in C^\gamma([0, T]; C^{1+\alpha}(\overline D)),\;b_i,\;c\in C^\gamma([0, T]; C^\alpha(\overline D))\quad (0< \alpha< 1), \]
\[ \sum^d_{i,j=1} a_{i,j}(t, x)\xi_i\xi_j\geq \mu|\xi|^2\text{ for all } \xi\in\mathbb{R}^d,\;\text{ for all }(t,x)\in [0, T]\times D\quad (\mu> 0). \] Then it is proved that for any \(f\in L^2(0,T; K(t)')\), (P) admits a weak solution \(u\in L^2(0,T; K(t))\), where \(K(t)= \{u\in H^1_0(\Omega); u|_{\partial{\mathcal O}(t)}= 0\}\), and \(K(t)'\) denotes the dual space of \(K(t)\). That is to say, \(u\) satisfies \[ \int^T_0 \{a(t; u(t),\phi(t))- (u(t), \phi'(t))_{L^2(D)}\} dt= \int^T_0\langle f(t),\phi(t)\rangle_{K(t)',K(t)}dt+ (u_0,\phi(0))_{L^2(D)} \] for all \(\phi\in L^2(0,T; K(t))\) with \(\phi_t\in L^2(0, T; L^2(D))\), \(\phi(T)= 0\). Here \[ a(t; u,v)= \int_D\left (\sum^d_{i,j=1} a_{i,j}(t, x)\partial_iu(x)\partial_j v(x)+ \sum^d_{i=1} b_i(t, x)\partial_i u(x) v(x)+ c(t,x) u(x) v(x)\right) dx. \] However, the uniqueness of the weak solution is open. This fact is assured by an abstract result due to J. L. Lions, a parabolic version of Lax-Milgram’s theorem, which is proved in the Appendix by using the elliptic regularization and Lax-Milgram’s theorem. In verifying the conditions required in this abstract result, the crucial point is to check the measurability of the projection \(P(t): K= H^1_0(D)\to K(t)\).
Furthermore, the interior regularity of weak solutions is discussed in Theorem 3.1, which says that if \(u_0\in C^{2+\beta}_0({\mathcal O}(0))\), \({\mathcal A}(0,\cdot)u_0+ f(0)\in C^{\beta+\gamma}_0({\mathcal O}(0))\) and \(f\in C^\gamma([0, T]; C^\beta(\overline D))\) with \(\beta\in (0,1/2)\), then for each point \((t_0, x_0)\in \Omega\), there exists \(\delta\), \(\mu> 0\) such that the weak solution \(u\) satisfies \(u\in C^{1+\gamma}([t_0- \delta,t_0+\delta]; C^\beta(\overline B(x_0; \mu)))\cap C^\gamma([t_0- \delta,t_0+\delta]; C^{2+\beta}(\overline B(x_0; \mu)))\).
The proof of this fact relies on the localization method and the smoothing effect of the analytic semigroup in \(L^p(D)\) generated by \({\mathcal A}\).

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
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