On nonlocal integro-differential boundary value problems for multidimensional pseudoparabolic equations (Q6641836)

The author studies the solvability of initial-boundary value problems for linear integro-differential equations, with a condition on the lateral boundary linking the values of the solution or the conormal derivative of the solution to the values of some integral operator of the solution. Theorems of existence and uniqueness of regular solutions are proved.\N\NLet \(\Omega\) be a bounded domain of space \(\mathbb{R}^n\) with a smooth (for simplicity, infinitely differentiable) boundary \(\Gamma\), \(Q\) be the cylinder \(\Omega \times (0, T)\) \((0 < T < +\infty)\), and \(S = \Gamma \times (0, T)\) be its lateral boundary. The functions \(a(x, t)\), \(c(x, t)\), and \(f(x, t)\) are defined in the cylinder \(\overline Q\), \(u_0(x)\) is defined in the domain \( \overline\Omega\), \(N(t)\) is defined in the interval \([0, T]\), and \(K_1(x, y, t)\), \(K_2(x, y, t)\) are defined for \(x \in \overline\Omega\), \(y \in \overline\Omega\), and \(t \in [0, T]\).\N\N{Boundary Value Problem I.} Find the function \(u(x, t)\), which is a solution in the cylinder \(Q\) of the equation \N\[\NLu \equiv \frac{\partial}{\partial t}(Au - \Delta u) - a(x, t) \Delta u + c(x, t) u = f(x, t), \]\N\[ Au = \int_0^t N(t - \tau) u(x, \tau) \, d\tau,\N\]\Nand such that the following conditions hold: \N\[\Nu(x, 0) = u_0(x), \quad x \in \Omega,\N\]\N\[\Nu(x, t)|_{(x,t) \in S} = \int_{\Omega} K_1(x, y, t) u(y, t) \, dy|_{(x,t) \in S} .\N\]\N\N{Boundary Value Problem II.} Find the function \(u(x, t)\), which is a solution in the cylinder \(Q\) of the equation\N\[\NLu \equiv \frac{\partial}{\partial t}(Au - \Delta u) - a(x, t) \Delta u + c(x, t) u = f(x, t), \]\N\[ Au = \int_0^t N(t - \tau) u(x, \tau) \, d\tau,\N\]\Nand such that the following conditions hold:\N\[\Nu(x, 0) = u_0(x), \quad x \in \Omega,\N\]\Nand the condition\N\[\N\frac{\partial u(x, t)}{\partial \nu(x)}\bigg|_{(x,t) \in S} = \int_{\Omega} K_2(x, y, t) u(y, t) \, dy|_{(x,t) \in S}\N\]\Nis satisfied.\N\NLet us give some prelimiaries. Define the operator \(M\) by the formula\N\[\N(Mu)(x, t) = u(x, t) - K u(x, t).\N\]\N\(M\) is a continuously invertible operator from \(L_2(\Omega)\) to \(L_2(\Omega)\), and there exist positive constants \(m_1, m_2\) such that the inequalities\N\[\Nm_1 \int_{\Omega} u^2(x, t) \, dx \leq \int_{\Omega} [Mu(x, t)]^2 \, dx \leq m_2 \int_{\Omega} u^2(x, t) \, dx \tag{1}\N\]\Nhold for any \(t \in [0, T]\) and \(u(x, t) \in L_\infty(0, T; L_2(\Omega))\). Let\N\[\NV = \{ v(x, t) : v \in L_\infty(0, T; W^2_2(\Omega)), \, v_t \in L_2(0, T; W^2_2(\Omega)) \}.\N\]\N\NThe following Poincaré inequality holds true:\N\[\N\int_0^t \int_{\Omega} u^2 \, dx \, d\tau \leq d_0 \int_0^t \int_{\Omega} (\Delta u)^2 \, dx \, d\tau,\N\]\Nwhere \( c_0, d_0 \) depend on the domain \(\Omega\).\N\NIntroduce the notations:\N\[\NP_0 = \max_{t \in [0,T]} \int_{\Omega} \int_{\Omega} (\Delta_x K_1)(x, y, t)^2 \, dx \, dy,\N\]\N\[\NQ_0 = \max_{t \in [0,T]} \int_{\Omega} \int_{\Omega} K_2^1(x, y, t) \, dx \, dy, \quad c_1 = \max_{t \in [0,T]} |N'(t)|.\N\]\N\NThen the following theorems are proved.\N\N{Theorem.} Suppose the conditions (1) are satisfied, and\N\[\Nc(x, t) \in C^1(\overline Q), \quad c(x, t) \geq c_0 > 0 \text { for } (x, t) \in \overline Q;\N\]\N\[\NK_1(x, y, t) \in C^3(\overline \Omega \times \overline \Omega \times [0, T]);\N\]\N\[\N1 - \delta_0^2(1 + d_0) - \frac{P_0 d_0}{\delta_0^2 m_1} > 0, \quad 1 - \frac{Q_0 d_0}{\delta_0^2 m_1} > 0 \text{ for } \delta_0 \in \left(0, \frac{\sqrt{2}}{2}\right);\N\]\Nfor \(f(x, t) \in L_2(Q).\)\N\NThen the boundary value problem I has a solution \(u(x, t)\) belonging to the space \(V\), and this solution is unique.\N\NLet us define the operator \(M_1\) as\N\[\N(M_1 u)(x, t) = u(x, t) - \int_\Omega K_2(x, y, t) u(y, t) \, dy.\N\]\NThe operator \(M_1\) is uniquely and continuously invertible as an operator from \(L_2(\Omega)\) to \(L_2(\Omega)\) for all \(t \in [0, T]\), and there exist positive constants \(m_3, m_4\) such that the following inequalities hold:\N\[\Nm_3 \int_{\Omega} u^2(x, t) \, dx \leq \int_{\Omega} [M_1 u(x, t)]^2 \, dx \leq m_4 \int_{\Omega} u^2(x, t) \, dx \tag{2}\N\]\Nfor any \(t \in [0, T]\) and \(u(x, t) \in L_\infty(0, T; L_2(\Omega))\).\N\N{Theorem.} Suppose the conditions (2) are satisfied, and\N\[\Na(x, t), c(x, t) \in C^1(\overline Q),\N\]\N\[\Na(x, t) \geq a_0 > 0, \quad c(x, t) \geq c_0 > 0 \text{ for } (x, t) \in \overline Q;\N\]\N\[\NK_2(x, y, t) \in C^3(\overline\Omega \times \overline\Omega \times [0, T]);\N\]\N\[\N1 - \delta_0^2(1 + d_0) - \frac{P_1 d_0}{\delta_0^2 m_3} > 0, \quad 1 - \frac{Q_1 d_0}{\delta_0^2 m_3} > 0 \text{ for } \delta_0 \in \left(0, \frac{\sqrt{2}}{2}\right);\N\]\Nfor \(f(x, t) \in L_2(Q)\), \(w_1(x) \in W^1_2(\Omega)\),\N\[\N\frac{\partial w_1(x)}{\partial \nu(x)}\bigg|_{x \in \Gamma} = 0.\N\]\NThen the boundary value problem II has a solution \(u(x, t)\).