Fujita-type blow-up for inhomogeneous semilinear heat equations with regularly varying forcing

Abstract: We develop a unified framework for Fujita-type blow-up of solutions to the inhomogeneous semilinear heat equation $$\partial_tu-Δu=|u|^{p+\mathbf{w}(x),} \qquad (t,x)\in(0,\infty)\times\mathbb{R}^N, \qquad u(0, \cdot)=u_0.$$ The classical integrability assumptions on the forcing term are replaced by quantitative regular variation properties of its spatial mass $F(R)=\int\limits_{|x|\le R}\mathbf{w}(x)\,dx.$ Using techniques from regular variation theory together with the Mitidieri--Pohozaev test-function method, we establish sharp Fujita-type nonexistence results and identify the critical exponent in terms of the variation index of $F$. We prove that global solutions do not exist in the subcritical range and obtain critical-case blow-up under suitable slowly varying corrections. The regular variation framework further shows the optimality of the underlying mass condition, extends naturally to anisotropic settings through operator regular variation, and yields sufficient blow-up criteria for sign-changing forcings via the Gaussian-Laplace transform. The approach also applies to space-time dependent forcings, Riesz-potential type forcings, and equations involving the fractional Laplacian, providing a unified description of blow-up thresholds beyond the classical Fujita theory.

• The paper establishes sharp blow-up criteria for inhomogeneous semilinear heat equations by replacing traditional integrability conditions with precise regular variation properties of the forcing mass.
• It extends classical Fujita results by incorporating anisotropic and operator regular variation to handle nonintegrable, sign-changing, and slowly decaying forcing terms.
• The unified Φ-formalism and detailed counterexamples offer critical insights into threshold exponents and the necessity of liminf lower bounds on the external forcing.

Fujita-Type Blow-Up for Inhomogeneous Semilinear Heat Equations with Regularly Varying Forcing

Introduction and Motivation

This paper (2606.31643) addresses the finite-time blow-up phenomenon for the inhomogeneous semilinear heat equation $$\partial_t u - \Delta u = |u|^p + \mathbf{w}(x), \quad (t,x) \in (0, \infty) \times \mathbb{R}^N, \quad u(0, x) = u_0(x),$$ with a broad class of spatially inhomogeneous or space-time dependent external forcings, focusing particularly on those whose spatial mass exhibits regular variation. The goal is to rigorously characterize the precise conditions under which no global nonnegative (weak) solution exists, extending classical Fujita-type results to inhomogeneous scenarios and non-integrable, anisotropic, or slowly decaying forcings.

Main Theoretical Contributions

Blow-Up Criteria via Regular Variation

The key innovation is the replacement of integrability conditions on the forcing $\mathbf{w}$ with sharp quantitative requirements on its spatial mass $F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$ Specifically, the authors employ regular variation theory: $F \in \mathrm{RV}_\gamma$ if $F(R) \sim R^{\gamma}L(R)$ for a slowly varying $L(R)$. The methodology allows the identification of the precise blow-up threshold $p_F(\gamma)$ determined by the variation index:

$$p_F(\gamma) = \begin{cases} \frac{N - \gamma}{N - \gamma - 2}, & \gamma < N - 2, \ \infty, & \gamma \geq N - 2. \end{cases}$$

The nonexistence of global weak solutions is shown for all $1 < p < p_F(\gamma)$ under minimal lower bounds on $F(R)$, including regularly varying (possibly nonintegrable) cases. At the critical exponent $\mathbf{w}$0, blow-up persists under subpolynomial amplification ($\mathbf{w}$1), reflecting the subtlety and sharpness of the regular variation framework.

Operator Regular Variation and Anisotropy

The analysis is extended to include anisotropic forcings via operator regular variation, incorporating dilation-invariant cones and exponent matrices $\mathbf{w}$2. The paper provides detailed blow-up criteria for situations in which the growth rates differ across directions, leading to a new family of critical exponents $\mathbf{w}$3 depending on the spectrum of $\mathbf{w}$4 and the asymptotic geometry of $\mathbf{w}$5.

Fine Analysis of Growth Hypotheses

A significant technical advancement is the clarification of the necessity of a $\mathbf{w}$6-type lower bound on the mass $\mathbf{w}$7. The authors construct examples demonstrating that replacing $\mathbf{w}$8 with $\mathbf{w}$9 is insufficient: there exist nonnegative forcings where $F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$0 has positive $F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$1 but zero $F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$2, eliminating the possibility of regular variation and invalidating blow-up arguments in the entire subcritical range.

Generalizations

The framework handles a broad spectrum of extensions, including:

• Space-time dependent forcing terms.
• Forcings given by Riesz potentials and operators of nonlocal type, specifically the fractional Laplacian $F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$3.
• Densities depending only on a subset of variables, with effective dimension reduction.
• Forcing terms that change sign, with precise Tauberian-type Laplace transform conditions linking the asymptotics of the spatially windowed mass to nonexistence.

Unified $F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$4-Formalism

The notion of cumulative-forcing functionals,

$F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$5

with $F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$6, serves to unify and clarify the blow-up thresholds in both space-time-inhomogeneous and nonlocal/fractional settings. The dichotomy $F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$7 for the index $F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$8 of $F(R) = \int_{|x| \le R} \mathbf{w}(x) \, dx.$9 against the corresponding upper-bound scaling $F \in \mathrm{RV}_\gamma$0 of the test-function method tightly characterizes subcritical and critical regimes.

Numerical Sharpness and Contradictory Results

The paper provides explicit families of forcings for which the new blow-up exponents strictly surpass the classical integrable ($F \in \mathrm{RV}_\gamma$1) case. For example, forcings with power-type decay $F \in \mathrm{RV}_\gamma$2 result in a strictly larger subcritical interval $F \in \mathrm{RV}_\gamma$3, while slow amplifications such as logarithmic corrections permit blow-up even at the classical Fujita threshold.

An example with highly sparse annular support demonstrates the possibility of $F \in \mathrm{RV}_\gamma$4 oscillating between large and vanishing mass, thus precluding regular variation and raising open questions about the minimal blow-up range under weak lower-mass conditions.

Furthermore, in explicit anisotropic weighting scenarios, the operator-RV blow-up range is shown to be strictly larger than both the isotropic and lower-dimensional projected versions, highlighting the importance of the geometric adaptation via operator regular variation.

Theoretical and Methodological Implications

These results systematically extend the canon of superlinear parabolic blow-up theory to include the widest documented class of inhomogeneous forcings and nonlocal operators. The utilization of Karamata theory, Tauberian theorems, and the Mitidieri–Pohozaev test function method underlines the interaction between scaling properties of the forcing mass and the nonlinear evolution.

Furthermore, anisotropic results address open questions about the optimality of thresholds when the geometry of the forcing is not isotropic or integrable, suggesting directions for refining both lower bounds and the effect of geometry in dispersive and dissipative PDEs.

The theoretical framework is also robust in handling sign-changing forcings with net positive mass, via transform-based (Laplace, Gaussian windows) criteria.

Speculation on Future Developments

Potential future investigations prompted by this work include:

• Determining the existence/nonexistence at the critical exponent with slowly varying factors that are bounded or decay.
• Optimization of blow-up exponents for general operator-scaling geometries, possibly maximizing over boxes not aligned to the exponent matrix.
• Removal or avoidance of one-sided Tauberian conditions for the treatment of sign-changing forcings.
• Extension to non-diagonal, non-expansive, or complex-exponent dilations and to coupled systems or other classes of parabolic or mixed local/nonlocal PDEs.
• Quantitative analysis of blow-up rates and lifespan asymptotics reflecting the precise regular variation behavior of the mass.

Conclusion

This paper (2606.31643) establishes a comprehensive and sharp characterization of finite-time blow-up in the inhomogeneous semilinear heat equation with a focus on regularly varying, potentially anisotropic and non-integrable, forcing terms. By embedding regular variation into the test-function blow-up paradigm, the theory encompasses and interpolates between classical discrete cases and irregular or slow-growth scenarios, yielding strong nonexistence ranges and precise criteria for criticality across a wide spectrum of inhomogeneous, nonlocal, and anisotropic settings. This framework paves the way for new research into the interplay of mass growth, operator geometry, and nonlinear evolution in parabolic PDEs.