On some parabolic integro-differential equations: existence and asymptotics of solutions.

*(English)*Zbl 0539.35074
Equadiff 82, Proc. int. Conf., Würzburg 1982, Lect. Notes Math. 1017, 161-167 (1983).

[For the entire collection see Zbl 0511.00014.]

Energy-type estimates and comparison methods range among the most widely used tools for investigating global existence and large-time asymptotics of solutions to semilinear parabolic differential equations. On the other hand, they are not immediately applicable to semilinear integro- differential equations. The paper under review indicates how energy like a-priori estimates can be used to establish existence and global asymptotic behavior of solutions to a class of parabolic integro- differential equations of delayed type \((*)\quad \partial_ tu-\Delta_ xu+a*g(u)=f\) in \(\Omega \times(0,T]; u=0\) in \(\partial \Omega \times(0,T];\) supplemented with the initial condition \(u(.,0)=h\). Here \(a*g(u)(x,t)=\int^{t}_{0}a(t-s)g(u(x,s))ds.\) Standard smoothness assumptions are made on the bounded space domain \(\Omega \subset R^ n\), whereas the regularity assumptions on a,g are quite weak (uniqueness of the solutions is in general not expected).

In first place, an existence result is presented. Its proof is based on the Leray-Schauder degree argument, and requires an a-priori estimate for u(.,t) in \(W^{2,2}(\Omega)\). This is achieved by energy methods, upon growth assumptions on g (depending on the space dimension n) and appropriate compatibility conditions on a, f, g (for example: g non- decreasing, \(a\geq 0\), a’\(\leq 0\), and \(\sup(q:a''(s)+qa'(s)>0\) for a.e. \(s)>0\). This estimate is based, in turn, on an inequality for a class of nonlinear Volterra integral operators, and implies certain decay properties of the kernel a. Extensions to the case of slowly decaying kernels are also considered. Further generalizations are also outlined, namely to systems, to the case in which the functional dependence arises through the boundary conditions, to a class of quasilinear cases, and to replacing the Laplacian with 2m-order elliptic operators.

Energy-type estimates and comparison methods range among the most widely used tools for investigating global existence and large-time asymptotics of solutions to semilinear parabolic differential equations. On the other hand, they are not immediately applicable to semilinear integro- differential equations. The paper under review indicates how energy like a-priori estimates can be used to establish existence and global asymptotic behavior of solutions to a class of parabolic integro- differential equations of delayed type \((*)\quad \partial_ tu-\Delta_ xu+a*g(u)=f\) in \(\Omega \times(0,T]; u=0\) in \(\partial \Omega \times(0,T];\) supplemented with the initial condition \(u(.,0)=h\). Here \(a*g(u)(x,t)=\int^{t}_{0}a(t-s)g(u(x,s))ds.\) Standard smoothness assumptions are made on the bounded space domain \(\Omega \subset R^ n\), whereas the regularity assumptions on a,g are quite weak (uniqueness of the solutions is in general not expected).

In first place, an existence result is presented. Its proof is based on the Leray-Schauder degree argument, and requires an a-priori estimate for u(.,t) in \(W^{2,2}(\Omega)\). This is achieved by energy methods, upon growth assumptions on g (depending on the space dimension n) and appropriate compatibility conditions on a, f, g (for example: g non- decreasing, \(a\geq 0\), a’\(\leq 0\), and \(\sup(q:a''(s)+qa'(s)>0\) for a.e. \(s)>0\). This estimate is based, in turn, on an inequality for a class of nonlinear Volterra integral operators, and implies certain decay properties of the kernel a. Extensions to the case of slowly decaying kernels are also considered. Further generalizations are also outlined, namely to systems, to the case in which the functional dependence arises through the boundary conditions, to a class of quasilinear cases, and to replacing the Laplacian with 2m-order elliptic operators.

Reviewer: P.de Mottoni

##### MSC:

35R10 | Functional partial differential equations |

35K55 | Nonlinear parabolic equations |

45K05 | Integro-partial differential equations |

34K99 | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

35B40 | Asymptotic behavior of solutions to PDEs |

35B45 | A priori estimates in context of PDEs |

35A05 | General existence and uniqueness theorems (PDE) (MSC2000) |