Finite Time Blow-up for a Non-linear Parabolic Equation with a Gradient Term and Applications

Dissipative Gradient Nonlinearities Prevent δ$\delta$‐Formations in Local and Nonlocal Attraction–Repulsion Chemotaxis Models

“…We similarly operate to derive (44) Finally, by using bounds ( 51) and ( 52) into relation (50) we obtain what claimed. □…”

Dissipative Gradient Nonlinearities Prevent δ$\delta$‐Formations in Local and Nonlocal Attraction–Repulsion Chemotaxis Models

“…We similarly operate to derive (44) Finally, by using bounds ( 51) and ( 52) into relation (50) we obtain what claimed. □…”

A Keller-Segel type taxis model with ecological interpretation and boundedness due to gradient nonlinearities

“…of the gradient of the population density, so that we obtain u t = ∆u−∇•(u∇v)+f (u)−g(∇u), where g is an increasing function of |∇u|. (Indeed, this coincides with the interpretation of gradient terms in scalar parabolic equations as 'accidental deaths' suggested by Souplet in [32].) 1.3.…”

“…IV]). Additionally, a possible biological interpretation of these terms was given in the above-mentioned [32]. Nevertheless, in chemotaxis models, gradient nonlinearities of this kind seem to have been treated only in [36], where a lower bound of the maximal existence time of solutions was derived.…”

Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term

“…This equation was first introduced by Chipot and Weissler [4] in 1989 with µ < 0 (and q > 1) as a model to understand whether blow-up may be prevented by the addition of the negative gradient term. Later in 1996, Souplet suggested a population dynamics interpretation for the equation in [25]. Many authors dealt with the mathematical analysis of this equation, both for large time dynamics and finite time blow-up, and also for the elliptic version (see [4,5,26,27,24,23,28,29,30] and references therein).…”

Dissipative Gradient Nonlinearities Prevent δ$\delta$‐Formations in Local and Nonlocal Attraction–Repulsion Chemotaxis Models

“…We similarly operate to derive (44) Finally, by using bounds ( 51) and ( 52) into relation (50) we obtain what claimed. □…”

A Keller-Segel type taxis model with ecological interpretation and boundedness due to gradient nonlinearities

“…of the gradient of the population density, so that we obtain u t = ∆u−∇•(u∇v)+f (u)−g(∇u), where g is an increasing function of |∇u|. (Indeed, this coincides with the interpretation of gradient terms in scalar parabolic equations as 'accidental deaths' suggested by Souplet in [32].) 1.3.…”

“…IV]). Additionally, a possible biological interpretation of these terms was given in the above-mentioned [32]. Nevertheless, in chemotaxis models, gradient nonlinearities of this kind seem to have been treated only in [36], where a lower bound of the maximal existence time of solutions was derived.…”

Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term

“…This equation was first introduced by Chipot and Weissler [4] in 1989 with µ < 0 (and q > 1) as a model to understand whether blow-up may be prevented by the addition of the negative gradient term. Later in 1996, Souplet suggested a population dynamics interpretation for the equation in [25]. Many authors dealt with the mathematical analysis of this equation, both for large time dynamics and finite time blow-up, and also for the elliptic version (see [4,5,26,27,24,23,28,29,30] and references therein).…”

Dissipative Gradient Nonlinearities Prevent δ$\delta$‐Formations in Local and Nonlocal Attraction–Repulsion Chemotaxis Models

“…We similarly operate to derive (44) Finally, by using bounds ( 51) and ( 52) into relation (50) we obtain what claimed. □…”

A Keller-Segel type taxis model with ecological interpretation and boundedness due to gradient nonlinearities

“…of the gradient of the population density, so that we obtain u t = ∆u−∇•(u∇v)+f (u)−g(∇u), where g is an increasing function of |∇u|. (Indeed, this coincides with the interpretation of gradient terms in scalar parabolic equations as 'accidental deaths' suggested by Souplet in [32].) 1.3.…”

“…IV]). Additionally, a possible biological interpretation of these terms was given in the above-mentioned [32]. Nevertheless, in chemotaxis models, gradient nonlinearities of this kind seem to have been treated only in [36], where a lower bound of the maximal existence time of solutions was derived.…”

Existence and stability of a blow-up solution with a new prescribed behavior for a heat equation with a critical nonlinear gradient term

“…This equation was first introduced by Chipot and Weissler [4] in 1989 with µ < 0 (and q > 1) as a model to understand whether blow-up may be prevented by the addition of the negative gradient term. Later in 1996, Souplet suggested a population dynamics interpretation for the equation in [25]. Many authors dealt with the mathematical analysis of this equation, both for large time dynamics and finite time blow-up, and also for the elliptic version (see [4,5,26,27,24,23,28,29,30] and references therein).…”