Liouville theorems and stability theorems for the solutions of parabolic systems

1964 ◽  
pp. 41-69
Author(s):  
S. D. Èĭdel’man
Keyword(s):  
Parabolic Systems ◽  
Liouville Theorems ◽  
Stability Theorems

scholarly journals A Note on Parabolic Liouville Theorems and Blow-Up Rates for a Higher-Order Semilinear Parabolic System

2011 ◽  
Vol 2011 ◽  
pp. 1-9 ◽  
Author(s):  
Guocai Cai ◽  
Hongjing Pan ◽  
Ruixiang Xing
Keyword(s):  
Parabolic System ◽  
Blow Up ◽  
Parabolic Systems ◽  
Time Variable ◽  
Liouville Type ◽  
Liouville Theorems ◽  
Semilinear Parabolic System ◽  
Liouville Type Theorems ◽  
Systems Of Inequalities ◽  
Semilinear Parabolic

We improve some results of Pan and Xing (2008) and extend the exponent range in Liouville-type theorems for some parabolic systems of inequalities with the time variable onR. As an immediate application of the parabolic Liouville-type theorems, the range of the exponent in blow-up rates for the corresponding systems is also improved.

Stability in 𝐿𝑞-norm for inverse source parabolic problems

2020 ◽  
Vol 28 (6) ◽  
pp. 797-814
Author(s):  
Elena-Alexandra Melnig
Keyword(s):  
Parabolic Equations ◽  
Main Tool ◽  
Parabolic Systems ◽  
Carleman Estimates ◽  
Parabolic Problems ◽  
Lebesgue Spaces ◽  
First Order ◽  
Lipschitz Estimates ◽  
Systems Of Parabolic Equations

AbstractWe consider systems of parabolic equations coupled in zero and first order terms. We establish Lipschitz estimates in {L^{q}}-norms, {2\leq q\leq\infty}, for the source in terms of the solution in a subdomain. The main tool is a family of appropriate Carleman estimates with general weights, in Lebesgue spaces, for nonhomogeneous parabolic systems.

scholarly journals Lorentz spaces in action on pressureless systems arising from models of collective behavior

2021 ◽  
Author(s):  
Raphaël Danchin ◽  
Piotr Bogusław Mucha ◽  
Patrick Tolksdorf
Keyword(s):  
Initial Data ◽  
Collective Behavior ◽  
Existence And Uniqueness ◽  
Maximal Regularity ◽  
Parabolic Systems ◽  
Lorentz Spaces ◽  
Regularity Estimate ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Time Existence

AbstractWe are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in $$L_{q,r}(0,T;L_p(\Omega ))$$ L q , r ( 0 , T ; L p ( Ω ) ) spaces.

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