Blow-up and non-extinction for a nonlocal parabolic equation with logarithmic nonlinearity

Abstract

This paper is devoted to studying a nonlocal parabolic equation with logarithmic nonlinearity u log |u| -ffl u log |u| dx in a bounded domain, subject to homogeneous Neumann boundary value condition. By using the logarithmic Sobolev inequality and energy estimate methods, we get the results under appropriate conditions on blow-up and non-extinction of the solutions, which extend some recent results.

Global Existence and Decay Estimates of Energy of Solutions for a New Class of p -Laplacian Heat Equations with Logarithmic Nonlinearity

“…So that, as u ∈ W 1,p 0 ðΩÞ \ f0g, we have d ≠ 0: And if u ∈ ℵ by (30), we obtain that λ * is the only critical point in ð0, ∞Þ of the mapping gðλÞ: Therefore,…”

“…If we read in recent research, we notice that logarithmic nonlinearity has been entered into nonrelativistic wave equations that describe spinning particles that move in an external electromagnetic field and in the relativistic wave equation for spinless particles (see, for example, [2,4,18,19]). In addition to what we mentioned above, this type of nonlinearity is used in various branches of physics such as optics, nuclear physics, geophysics, and inflationary cosmology (to read about this in detail, see [18][19][20][21][22][23][24][25][26][27][28][29][30][31]). Given all the basic previous meanings in physics, the study of universal solutions of this type of nonlinear logarithms is of great interest on the part of mathematicians.…”

Qualitative analysis of solutions for the p‐Laplacian hyperbolic equation with logarithmic nonlinearity

“…[1][2][3]13 This type of nonlinearity was introduced in the nonrelativistic wave equations describing spinning particles moving in an external electromagnetic field and also in the relativistic wave equation for spinless particles. 4,30,31 Moreover, the logarithmic nonlinearity appears in several branches of physics such as inflationary cosmology, 5 nuclear physics, 6 optics, 7 and geophysics. 8 With all those specific underlying meaning in physics, the global-in-time well-posedness of solution to the problem of evolution equation with such logarithmic type nonlinearity captures lots of attention.…”

Global Existence and Decay of Solutions for a Class of a Pseudo-Parabolic Equation with Singular Potential and Logarithmic Nonlocal Source

“…Interested individuals can read reference materials [4] [5] [6]. Yan et al [7] considered the following parabolic equation:…”