Parabolic Abstract Evolution Equations

• Parabolic abstract evolution equations are PDE systems with longitudinal diffusion and transverse dynamics in cylindrical geometries.
• The framework utilizes uniformly local Sobolev spaces and integrated semigroup theory to address challenges from non-dense domains and non-sectorial operators.
• These equations play a crucial role in modeling pattern formation and front propagation in systems exhibiting infinite local energy.

A parabolic abstract evolution equation is a partial differential equation (PDE) system, typically of the form

$$\partial_t u(x, t) = \Delta u(x, t) - B u(x, t) + F(u(x, t)),$$

where $u$ takes values in a Hilbert space $Y$, $\Delta$ is a Laplacian in one or more “longitudinal” spatial variables (e.g., $x \in \mathbb{R}^d$), $B$ is a positive self-adjoint operator on $Y$ dictating “transverse” diffusion, and $F$ is a (possibly nonlinear) reaction term. The analysis in cylindrical domains and uniformly local Sobolev spaces centers on understanding these systems when classical energy methods may fail, especially for solutions with infinite total energy but locally finite energy, and for operators with non-dense domain in the underlying function space (Romain, 7 Aug 2025).

1. Model Equation and Cylindrical Geometry

The primary class of equations is

$$\partial_t u(x, t) = \Delta u(x, t) - B u(x, t) + F(u(x, t)),$$

where:

• $x \in \mathbb{R}^d$ (“longitudinal variable”), $u(x, t) \in Y$, and the Laplacian acts in the $x$-variable.
• $B$ acts in the transverse Hilbert space $Y$. For instance, if $Y = L^2(\omega)$, with $\omega \subset \mathbb{R}^{d'}$ a bounded domain, then $B = -\Delta_y$ (possibly with Dirichlet/Neumann boundary conditions), so $u: \mathbb{R}^d \times \omega \to \mathbb{R}$ and the system naturally describes PDEs on $\mathbb{R}^d \times \omega$ (“cylindrical domains”).
• $F(\cdot)$ is a (possibly nonlinear) reaction term, which may act pointwise, e.g., as a polynomial nonlinearity for scalar-valued $u$.

This setting arises naturally in the paper of front propagation, pattern formation, and other phenomena where solutions do not decay as $|x| \to \infty$.

2. Role of the Transverse Hilbert Space and Operator $B$

The introduction of a nontrivial Hilbert space $Y$ for the values of $u$ leads to an anisotropic structure:

• Diffusion along the longitudinal direction is given by the Laplacian $\Delta$.
• Diffusion in the transverse direction is encoded in the self-adjoint operator $B$.
• In concrete settings such as $\mathbb{R} \times \omega$, $u$ encodes both the longitudinal behavior (e.g., wave propagation or invasion direction) and localized energy in the cross-section.

The operator $B$ may be unbounded—for example, $B = -\Delta_y$ on $L^2(\omega)$—which has consequences for the function space theory, particularly with regard to the density of its domain in uniformly local spaces.

3. Uniformly Local (ul) Spaces and Infinite Energy

The function spaces of interest are the uniformly local $L^2$ and Sobolev spaces, defined as

$$\|u\|_{L^2_{\mathrm{ul}}(\mathbb{R}^d, Y)} := \sup_{a \in \mathbb{R}^d} \|u\|_{L^2(B(a,1), Y)},$$

where $B(a,1)$ is the ball of radius 1 centered at $a$.

Two versions are crucial:

• Weak $L^2_{\mathrm{ul}\text{-}w}$: functions with finite ul-norm, no continuity under translation required.
• Strong $L^2_{\mathrm{ul}\text{-}s}$: functions with finite ul-norm and uniformly continuous under translation (i.e., the map $\xi \mapsto u(\cdot) - u(\cdot - \xi)$ is continuous in the ul-norm).

These spaces are designed to capture solutions with infinite total energy but locally finite energy everywhere (e.g., invasion fronts, spatially periodic or random patterns).

4. Cauchy Problem Formulation and Well-posedness in Non-Dense Domains

A central analytic challenge is that the natural realization of the operator $A = -\partial_{xx} + B$ on $L^2_{\mathrm{ul}}(\mathbb{R}^d, Y)$,

$$D(A) = H^2_{\mathrm{ul}}(\mathbb{R}^d, Y) \cap L^2_{\mathrm{ul}}(\mathbb{R}^d, D(B)),$$

may not be a dense subspace unless $B$ is bounded or $Y$ is finite dimensional.

• In the scalar case ($Y = \mathbb{R}$), it is known that the Cauchy problem is ill-posed in the weak space but well-posed in the strong space, thanks to uniform continuity ensuring density and thus sectoriality of $A$.
• For general $Y$ and especially unbounded $B$, neither the weak nor the strong versions guarantee density of $D(A)$ in the ambient space.
• The lack of domain density means that $A$ does not generate a strongly continuous semigroup; the standard theory of analytic or sectorial operators is not applicable.

To address well-posedness, the theory uses integrated semigroups (as in Sinestrari and Da Prato), which provide a framework for defining solutions when the generator's domain is not dense. Solutions constructed via this approach are “mild” in the sense of the integrated semigroup theory and remain meaningful even in the absence of strong continuity.

5. Sectoriality and Its Failure in Uniformly Local Spaces

Sectoriality is a sufficient condition for the existence of analytic semigroups and is typically assured by density and elliptic regularity. In uniformly local Lebesgue spaces:

• When $B$ is bounded, the spectrum of $A$ is not an issue, and density can be recovered in the strong ul-space.
• When $B$ is unbounded, density fails for both ul-space versions, so $-\partial_{xx} + B$ is not sectorial. The example provided in (Romain, 7 Aug 2025) is not artificial; this failure reflects features of many non-compact or cylindrical PDE problems of physical interest (e.g., reaction-diffusion equations on $\mathbb{R} \times \omega$ with Dirichlet or Neumann cross-sectional boundary conditions).

This suggests that non-sectorial operators with non-dense domain naturally arise in infinite energy settings and that classical methods must be replaced by abstract parabolic operator theory.

6. Comparison of Weak and Strong Uniformly Local Spaces

While in the scalar case strong ul-spaces compensate for the pathology of weak ul-spaces, for $Y$-valued functions (with unbounded $B$), both spaces yield non-dense domains and hence non-sectorial generators. The distinguishing features observed in lower-dimensional, scalar settings are shown not to generalize; both space versions behave similarly with respect to generator properties in the presence of nontrivial $B$.

A plausible implication is that for anisotropic, infinite energy settings, any attempt to use classical “stronger” function space topologies for regularization must be informed by the underlying operator structure, not just the translation properties of the norm.

7. Non-Dense Differential Operators: A Canonical Example

The operator

$$A = -\partial_{xx} + B \quad \text{on} \quad D(A) = H^2_{\mathrm{ul}}(\mathbb{R}, Y) \cap L^2_{\mathrm{ul}}(\mathbb{R}, D(B))$$

on $L^2_{\mathrm{ul}}(\mathbb{R}, Y)$ (both weak/strong versions) provides a “natural” example of a differential operator with non-dense domain. The essential source of non-density is the absence of sufficiently many smooth, compactly supported functions (which are typically used for density in Sobolev space theory) in the uniformly local context for infinite-dimensional $Y$ with compact embedding $D(B) \subset Y$.

This operator, which is directly relevant for cylindrical and anisotropic nonlinear parabolic problems, demonstrates that the integrated semigroup theory is essential for constructing solutions and that non-dense domains are an inherent feature of the analytic framework for these PDE classes.

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The work broadens the scope of parabolic abstract evolution equations to accommodate infinite energy, anisotropic/cylindrical geometries, and operator-theoretic pathologies that arise in physically realistic unbounded or infinite domains (Romain, 7 Aug 2025). The development provides robust analytical techniques for investigating solution behavior—even in weakly regular or noncompact settings—through abstract functional analysis and generalized semigroup theory.