The Fujita exponent across an interface

Technical Summary: The Fujita Exponent Across an Interface

Problem Statement
This paper investigates the Cauchy problem for the semilinear parabolic equation:
$$\partial_t u = \mathcal{L} u + |u|^{p-1}u, \quad \text{in } (0, \infty) \times \mathbb{R}^N,$$
with initial data $u(0, x) = u_0(x)$. The operator $\mathcal{L}$ is defined as $\mathcal{L} = \Delta + 2q \delta_S \nabla$, where $\Delta$ is the standard Laplacian, $S = \{x \in \mathbb{R}^N \mid x_N = 0\}$ is a fixed interface hyperplane, $\delta_S$ is the surface distribution supported on $S$, and $q \in [-1, 1]$ is a real parameter.

The operator $\mathcal{L}$ acts as the standard Laplacian in the half-spaces $\{x_N > 0\}$ and $\{x_N < 0\}$ but imposes a singular drift on $S$. This results in a transmission condition on the normal derivative for smooth functions $f$ in the domain of $\mathcal{L}$:
$$(1-q) \partial_{x_N} f(\tilde{x}, 0^+) = (1+q) \partial_{x_N} f(\tilde{x}, 0^-), \quad \tilde{x} \in \mathbb{R}^{N-1}.$$
Probabilistically, this operator generates a skew Brownian motion where the parameter $q$ controls the asymmetry of crossing the interface. The central question addressed is whether the presence of this singular, non-homogeneous interface drift modifies the classical Fujita critical exponent $p_F = 1 + 2/N$, which governs the dichotomy between finite-time blow-up and global existence for the standard semilinear heat equation.

Methodology
The analysis relies on a combination of probabilistic interpretations and analytic estimates:

1. Fundamental Solution and Kernel Estimates: The authors utilize the explicit representation of the fundamental solution $G_q(x, y, t)$ derived from the theory of skew Brownian motion. A crucial step is establishing two-sided Gaussian bounds for $G_q$:
   $$(1-|q|) g(x-y, t) \leq G_q(x, y, t) \leq (1+|q|) g(x-y, t),$$
   where $g$ is the standard heat kernel. This comparison ensures that the large-scale diffusive behavior of $\mathcal{L}$ is indistinguishable from the Laplacian, despite the lack of spatial homogeneity.

2. Local Well-Posedness (Lebesgue Spaces): The existence and uniqueness of mild solutions are established using a contraction mapping argument in weighted Lebesgue spaces. The authors adapt the standard approach to account for the lack of translation invariance by exploiting the pointwise Gaussian bounds of $G_q$ to derive $L^q-L^r$ smoothing estimates identical in scaling to the classical heat semigroup. The analysis distinguishes between the supercritical regime ($q > N(p-1)/2$) and the critical endpoint ($q = N(p-1)/2$), employing auxiliary exponents and smallness conditions on the linear flow for the latter.

3. Blow-Up Analysis (Test-Function Method): To prove non-existence of global solutions for $p \leq p_F$, the authors employ the test-function method. A key technical innovation is the construction of a test function $\psi(t, x)$ that is even in the normal variable $x_N$ (i.e., $\psi(t, \tilde{x}, x_N) = \psi(t, \tilde{x}, -x_N)$). This symmetry ensures that the normal derivative $\partial_{x_N} \psi$ vanishes on the interface $S$, causing the singular interface term in the weak formulation to vanish. Consequently, the weak formulation reduces to an identity identical to the classical case, allowing the standard Fujita argument to proceed.

4. Global Existence: For $p > p_F$, global existence for small initial data is proven using a fixed-point argument in a Banach space of functions with specific time-weighted norms, leveraging the established smoothing properties of the semigroup.

Key Contributions and Results

• Local Well-Posedness: The paper establishes the local well-posedness of mild solutions in $L^q(\mathbb{R}^N)$ for $q \geq 1$ (with specific conditions on $q$ relative to $p$). It provides a functional framework that tracks the smoothing effects away from the interface and the loss of scaling invariance induced by the transmission structure.
• Sharp Fujita Dichotomy: The main result is the proof that the critical Fujita exponent for this interface problem remains exactly $p_F = 1 + 2/N$.
  • Blow-Up: If $1 < p \leq 1 + 2/N$ and the initial data $u_0 \geq 0$ is non-trivial (with positive total mass), every non-negative weak solution blows up in finite time.
  • Global Existence: If $p > 1 + 2/N$, global-in-time mild solutions exist for sufficiently small initial data in $L^{q_c}(\mathbb{R}^N)$, where $q_c = N(p-1)/2$.
• Robustness of the Exponent: The study demonstrates that the singular interface drift, while breaking spatial homogeneity and scaling invariance, does not alter the fundamental scaling exponents governing the blow-up phenomenon. The interface affects the constants in the kernel estimates but preserves the critical threshold.

Significance
The paper claims to provide the first result of this type for operators involving a singular drift supported on a hypersurface. The significance lies in showing that the Fujita phenomenon is stable under the presence of discontinuous diffusion effects and interface transmission conditions. The critical exponent is intrinsic to the equation's diffusive scaling, which remains governed by the Gaussian decay of the fundamental solution, even in the presence of the skew interface. The results extend the classical theory of semilinear heat equations to a class of non-homogeneous operators arising in the study of skew Brownian motion and diffusion in heterogeneous media.

Limitations and Open Problems
The authors note that the "doubly critical" case where $q = N(p-1)/2 = 1$ remains open. Furthermore, the paper outlines several directions for future research, including:

• Extending the results to curved interfaces or higher-codimension surfaces, where tangential-normal factorization fails.
• Investigating variable skewness parameters $q(\tilde{x})$ or state-dependent transmission conditions.
• Analyzing fractional diffusion operators with interface drifts.
• Studying multiple interfaces or time-dependent/random interfaces.