Manuscript Topics

An evolution equation is a partial differential equation that describes the time evolution of a physical system starting from given initial data. Evolution equations arise from many areas of applied and engineering sciences. For example, nonlinear Schrödinger equations from quantum mechanics, Navier-Stokes equations from fluid mechanics, nonlinear reaction-diffusion equations from heat transfers and biological sciences, and Korteweg–de Vries equation from water-waves theory, to name just a few, are special examples of nonlinear evolution equations.

Starting with the seminal works of Kaplan and Fujita, a great number of papers and techniques concerning blowing-up solutions of evolution equations of various types appeared during the last sixty years. In 1995, Sugitani initiated blow-up solutions of evolution equations with non-local in space operators, while in 2005, Kirane and his collaborators initiated studies on blow-up for time-fractional evolution equations. Although papers on blow-up for evolution equation are appearing at a noticeable speed, there is still a need of qualitative and numerical studies.

The objective of this Special Issue is to report on the cutting edge development of the blow-up for nonlinear evolution equations and their applications. The Special issue will bring together experts in qualitative study of evolution equations, numerical analysts, and practitioners in the various applied fields of contemporary natural sciences where blow-up reveals either a genuine phenomenon or a limitation of the models. The results on solution blow-up phenomena and its numerical treatment will be reported. The topic issue will focus on but not limited to:

• Finite-time blow-up for nonlinear time-fractional evolution equations
• Fujita-type results and critical exponents
• Instantaneous blow-up
• Evolution equations with singular coefficients
• Decay of mass for nonlinear evolution equations
• Estimates of lifespan and blow-up rates
• Numerical methods for estimating the blow-up time
• Numerical methods for solving nonlinear time-fractional evolution equations

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