# American Institute of Mathematical Sciences

• Previous Article
On boundary regularity of solutions to Maxwell's equations with a homogeneous conservative boundary condition
• DCDS-S Home
• This Issue
• Next Article
Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction
September  2009, 2(3): 449-471. doi: 10.3934/dcdss.2009.2.449

## Singularly non-autonomous semilinear parabolic problems with critical exponents

 1 Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Caixa postal 668, 13560-970 São Carlos, São Paulo, Brazil 2 Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos SP, Brazil

Received  September 2008 Revised  December 2008 Published  June 2009

In this work we consider initial value problems of the form

$\frac{dx}{dt} + A(t)x = f(t,x)$
$x(\tau)=x_0,$

in a Banach space $X$ where $A(t):D\subset X\to X$ is a linear, closed and unbounded operator which is sectorial for each $t$. We show local well posedness for the case when the nonlinearity $f$ grows critically. Applications to semilinear parabolic equations and strongly damped wave equations are considered.

Citation: Alexandre Nolasco de Carvalho, Marcelo J. D. Nascimento. Singularly non-autonomous semilinear parabolic problems with critical exponents. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 449-471. doi: 10.3934/dcdss.2009.2.449
 [1] Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543 [2] Mahesh G. Nerurkar. Spectral and stability questions concerning evolution of non-autonomous linear systems. Conference Publications, 2001, 2001 (Special) : 270-275. doi: 10.3934/proc.2001.2001.270 [3] Xianlong Fu. Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evolution Equations & Control Theory, 2017, 6 (4) : 517-534. doi: 10.3934/eect.2017026 [4] Yanan Li, Alexandre N. Carvalho, Tito L. M. Luna, Estefani M. Moreira. A non-autonomous bifurcation problem for a non-local scalar one-dimensional parabolic equation. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5181-5196. doi: 10.3934/cpaa.2020232 [5] Pengyu Chen, Xuping Zhang. Non-autonomous stochastic evolution equations of parabolic type with nonlocal initial conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4681-4695. doi: 10.3934/dcdsb.2020308 [6] Cung The Anh, Le Van Hieu, Nguyen Thieu Huy. Inertial manifolds for a class of non-autonomous semilinear parabolic equations with finite delay. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 483-503. doi: 10.3934/dcds.2013.33.483 [7] Mustapha Mokhtar-Kharroubi. On permanent regimes for non-autonomous linear evolution equations in Banach spaces with applications to transport theory. Kinetic & Related Models, 2010, 3 (3) : 473-499. doi: 10.3934/krm.2010.3.473 [8] Lu Yang, Meihua Yang, Peter E. Kloeden. Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2635-2651. doi: 10.3934/dcdsb.2012.17.2635 [9] Hannes Meinlschmidt, Joachim Rehberg. Hölder-estimates for non-autonomous parabolic problems with rough data. Evolution Equations & Control Theory, 2016, 5 (1) : 147-184. doi: 10.3934/eect.2016.5.147 [10] Aníbal Rodríguez-Bernal, Alejandro Vidal–López. Existence, uniqueness and attractivity properties of positive complete trajectories for non-autonomous reaction-diffusion problems. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 537-567. doi: 10.3934/dcds.2007.18.537 [11] Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 [12] K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038 [13] Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211 [14] Pengyu Chen, Xuping Zhang. Approximate controllability of nonlocal problem for non-autonomous stochastic evolution equations. Evolution Equations & Control Theory, 2021, 10 (3) : 471-489. doi: 10.3934/eect.2020076 [15] Shengfan Zhou, Linshan Wang. Kernel sections for damped non-autonomous wave equations with critical exponent. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 399-412. doi: 10.3934/dcds.2003.9.399 [16] Tôn Việt Tạ. Existence results for linear evolution equations of parabolic type. Communications on Pure & Applied Analysis, 2018, 17 (3) : 751-785. doi: 10.3934/cpaa.2018039 [17] Julia García-Luengo, Pedro Marín-Rubio, José Real, James C. Robinson. Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 203-227. doi: 10.3934/dcds.2014.34.203 [18] Dingshi Li, Xuemin Wang. Regular random attractors for non-autonomous stochastic reaction-diffusion equations on thin domains. Electronic Research Archive, 2021, 29 (2) : 1969-1990. doi: 10.3934/era.2020100 [19] Roberta Fabbri, Russell Johnson, Carmen Núñez. On the Yakubovich frequency theorem for linear non-autonomous control processes. Discrete & Continuous Dynamical Systems, 2003, 9 (3) : 677-704. doi: 10.3934/dcds.2003.9.677 [20] Serena Dipierro, Enrico Valdinoci. (Non)local and (non)linear free boundary problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 465-476. doi: 10.3934/dcdss.2018025

2020 Impact Factor: 2.425