Existence, uniqueness and blow-up of solutions for generalized auto-convolution Volterra integral equations

Mostafazadeh M., Shahmorad S., Erdoğan F.

Applied Mathematics and Computation, vol.471, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 471
  • Publication Date: 2024
  • Doi Number: 10.1016/j.amc.2024.128608
  • Journal Name: Applied Mathematics and Computation
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Compendex, Computer & Applied Sciences, INSPEC, Public Affairs Index, zbMATH, Civil Engineering Abstracts
  • Keywords: Auto-convolution, Existence, Finite-time blow-up, Uniqueness, Volterra integral equation
  • Van Yüzüncü Yıl University Affiliated: Yes

Abstract

In this paper, our intention is to investigate the blow-up theory for generalized auto-convolution Volterra integral equations (AVIEs). To accomplish this, we will consider certain conditions on the main equation. This will establish a framework for our analysis, ensuring that the solution of the equation exists uniquely and is positive. Firstly, we analyze the existence and uniqueness of a local solution for a more general class of AVIEs (including the proposed equation in this paper) under certain hypotheses. Subsequently, we demonstrate the conditions under which this local solution blows up at a finite time. In other words, the solution becomes unbounded at that time. Furthermore, we establish that this blow-up solution can be extended to an arbitrary interval on the non-negative real line, thus referred to as a global solution. These results are also discussed for a special case of generalized AVIEs in which the kernel functions are taken as positive constants.