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<p>In this paper, we study the global existence and blow-up phenomena of the nonlocal boundary value problem for porous medium equation with a nonlinear memory term. By virtue of Kaplan’s technique and some new properties on the system of differential inequalities, the method of constructing global existence upper solution and blow-up subsolutions, respectively, we establish sufficient conditions to guarantee global existence and finite-time blow-up for the solution; moreover, we give the upper bounds of blow-up time. Then, based on constructing blow-up supersolutions and combining the auxiliary function method with the modified differential inequality techniques, lower bounds of blow-up time are derived. Finally, we provide two concrete examples to illustrate our findings. Moreover, we use the finite difference method to numerically simulate the equations and present figures for visual observation of the blow-up phenomenon.</p>

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Blow-up and global analysis of solutions of porous medium equation with a nonlinear memory term

  • 期刊论文

  • 发表日期：

    2025年6月8日

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  • Tao Liu,
  • Lingling Zhang,
  • Hongwei Liu

摘要

In this paper, we study the global existence and blow-up phenomena of the nonlocal boundary value problem for porous medium equation with a nonlinear memory term. By virtue of Kaplan’s technique and some new properties on the system of differential inequalities, the method of constructing global existence upper solution and blow-up subsolutions, respectively, we establish sufficient conditions to guarantee global existence and finite-time blow-up for the solution; moreover, we give the upper bounds of blow-up time. Then, based on constructing blow-up supersolutions and combining the auxiliary function method with the modified differential inequality techniques, lower bounds of blow-up time are derived. Finally, we provide two concrete examples to illustrate our findings. Moreover, we use the finite difference method to numerically simulate the equations and present figures for visual observation of the blow-up phenomenon.