Parabolic abstract evolution equations in cylindrical domains and uniformly local Sobolev spaces
Abstract
In this article, we consider parabolic equations of the form ∂t u(x,t)=Δ u(x,t) − Bu(x,t) + F(u(x,t)) where u is valued in a transverse Hilbert space Y and B is a positive self-adjoint operator on Y, which allows for a different diffusion mechanism in the transverse direction. Our aim is to consider solutions with infinite energy and we will study the Cauchy problem in the uniformly local spaces associated with the norm |u|L2ul(RR,Y)= supa∈RR |u|L2(B(a,1),Y). For the classical parabolic equation, i.e. when Y=RR, it is known that the Cauchy problem is ill-posed in the weak version of the uniformly local spaces but well-posed in a stronger version, where additional uniform continuity is required. In this paper, we show that the linear operator ∂2xx − B is not necessarily a sectorial operator in any version of the uniformly local Lebesgue space, due to the possible non-density of its domain. We then use the theory of parabolic abstract evolution equations to formulate a well-posed Cauchy problem, even in the weak version of the uniformly local space. In particular, we believe that this paper offers a new perspective on the comparison between the two versions of the uniformly local spaces and also provides a new natural example of differential operators with non-dense domain.
Related
- Well-posedness and exponential stability for abstract evolution equations with delay in the nonlinear source: frictional and viscoelastic cases
- On the derivative loss for a class of abstract evolution equations with time dependent oscillating coefficients vanishing at initial time
- Banach-Valued Stieltjes Calculus and Linear Parabolic Evolution Equations with Posterior States
- Reconstruction of solutions to abstract Schrodinger, diffusion, and wave-type equations from snapshots
- Local and global C1,β-regularity for uniformly elliptic quasilinear equations of p-Laplace and Orlicz-Laplace type
BibTeX
@article{joly2026parabolic,
title = {{Parabolic abstract evolution equations in cylindrical domains and uniformly local Sobolev spaces}},
author = {Romain Joly},
journal = {Annales Mathématiques Blaise Pascal},
year = {2026},
month = oct,
url = {https://hal.science/hal-05830103v1},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Parabolic abstract evolution equations in cylindrical domains and uniformly local Sobolev spaces
AU - Romain Joly
PY - 2026
DA - 2026/10/21
JO - Annales Mathématiques Blaise Pascal
UR - https://hal.science/hal-05830103v1
AB - In this article, we consider parabolic equations of the form ∂_t u(x,t)=Δ u(x,t) − Bu(x,t) + F(u(x,t)) where u is valued in a transverse Hilbert space Y and B is a positive self-adjoint operator on Y, which allows for a different diffusion mechanism in the transverse direction. Our aim is to consider solutions with infinite energy and we will study the Cauchy problem in the uniformly local spaces associated with the norm |u|_L^2_ul(RR,Y)= sup_a∈RR |u|_L^2(B(a,1),Y). For the classical parabolic equation, i.e. when Y=RR, it is known that the Cauchy problem is ill-posed in the weak version of the uniformly local spaces but well-posed in a stronger version, where additional uniform continuity is required. In this paper, we show that the linear operator ∂^2_xx − B is not necessarily a sectorial operator in any version of the uniformly local Lebesgue space, due to the possible non-density of its domain. We then use the theory of parabolic abstract evolution equations to formulate a well-posed Cauchy problem, even in the weak version of the uniformly local space. In particular, we believe that this paper offers a new perspective on the comparison between the two versions of the uniformly local spaces and also provides a new natural example of differential operators with non-dense domain.
SN - 1259-1734
SN - 2118-7436
ER -