Numerical analysis of finite-time blow-up in nonlinear integro-differential equations

Volterra型非线性积分方程组出现在固体可燃材料热点火的数学模型中(例如，在钢的剪切带的形成中，当承受非常高的应变率时；见C.A.Roberts在J.Comput中的评论论文。APPL数学205(2007)，736-743及其参考文献)。这类方程--其解在有限时间内爆破--也与反应扩散类型的半线性抛物型偏微分方程有关；比较J.Bebernes和D.Eberly的《燃烧理论中的数学问题》(1989年)一书，以及H.A.Levine在SIAM Review 32(1990)上的评论论文，262-288。 在这个方案中，我们将研究具有爆破解的非线性Volterra方程的数值分析和计算解，重点是计算爆破时间和得到实际的误差估计。由于在许多应用中，解是否在有限时间内爆破或是否全局存在是先验未知的，因此我们将特别讨论用数值方法检测爆破的重要问题。这对于非线性Volterra积分方程和半线性抛物型积分-微分方程(包含Volterra记忆项形式的非局部反应项)都是这样做的。在后一种情况下，当基本空间域是无界的(例如扇形或锥形区域)时，有限时间爆破理论基本上是开放的。该提议还将试图弥合这一差距，因为对这一理论(例如爆破点的位置)的透彻理解将是为此类非局部爆破问题设计有效的数值格式的关键。

Nonlinear integral equations of Volterra type arise in the mathematical modelling of thermal ignition in solid combustible materials (e.g. in the formation of shear bands in steel, when subjected to very high strain rates; see the review paper by C.A. Roberts in J. Comput. Appl. Math 205 (2007), 736-743, and its references). Such equations -- whose solutions blow up in finite time -- are also related to semilinear parabolic partial differential equations of reaction-diffusion type; compare the book 'Mathematical Problems from Combustion Theory' (1989) by J. Bebernes and D. Eberly, and the review paper by H.A. Levine in SIAM Review 32 (1990), 262-288. In this proposal we shall study the numerical analysis and computational solution of nonlinear Volterra equations with blow-up solutions, with the focus being on computing the blow-up time and deriving realistic error estimates. Since in many applications it is not known a priori if the solution will blow up in finite time or if it exists globally, we shall in particular address the important problem of detecting blow-up numerically. This will be done both for nonlinear Volterra integral equations and for semilinear parabolic integro-differential equations (containing nonlocal reaction terms in the form of Volterra memory terms)). In te latter case the theory of finite-time blow-up when the underlying spatial domain is unbounded (e.g. a sectorial or cone-like domain) is essentially open. The proposal will also try to close this gap, since a thorough understanding of this theory (e.g. the location of the blow-up points) will be the key for designing effective numerical schemes for such nonlocal blow-up problems.