Nonexistence of solutions for singular quasilinear differential inequalities with a gradient nonlinearity

2012 ◽  
Vol 75 (5) ◽  
pp. 2812-2822 ◽  
Author(s):  
Xiaohong Li ◽  
Fengquan Li
Keyword(s):  
Differential Inequalities ◽  
Nonexistence Of Solutions ◽  
Gradient Nonlinearity

scholarly journals New Fujita type results for quasilinear parabolic differential inequalities with gradient dissipation terms

2021 ◽  
Vol 6 (10) ◽  
pp. 11482-11493
Author(s):  
Xiaomin Wang ◽  
◽  
Zhong Bo Fang
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Critical Exponents ◽  
Differential Inequality ◽  
Source Term ◽  
Differential Inequalities ◽  
Dissipation Term ◽  
Slow Decay ◽  
Nonexistence Of Solutions ◽  
Quasilinear Parabolic

<abstract><p>This paper deals with the new Fujita type results for Cauchy problem of a quasilinear parabolic differential inequality with both a source term and a gradient dissipation term. Specially, nonnegative weights may be singular or degenerate. Under the assumption of slow decay on initial data, we prove the existence of second critical exponents $ \mu^{*} $, such that the nonexistence of solutions for the inequality occurs when $ \mu &lt; \mu^{*} $.</p></abstract>

scholarly journals Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

2021 ◽  
Vol 143 (2) ◽  
pp. 301-335
Author(s):  
Jendrik Voss ◽  
Ionel-Dumitrel Ghiba ◽  
Robert J. Martin ◽  
Patrizio Neff
Keyword(s):  
Differential Inequalities ◽  
Two Dimensional ◽  
Ellipticity Condition ◽  
One Dimensional ◽  
Suitable Sense ◽  
Precise Analysis ◽  
Rank One ◽  
Isotropic Elasticity ◽  
Convexity Conditions

AbstractWe consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\frac{\mu }{2} \hspace{0.07em} \frac{\lVert F \rVert ^{2}}{\det F}+f(\det F)$ W ( F ) = μ 2 ∥ F ∥ 2 det F + f ( det F ) ; such an energy is rank-one convex if and only if the function $f$ f is convex.

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