The Fujita exponent across an interface

Abstract: We consider the semilinear parabolic equation [ \partial_t u = Δu + 2\mathfrak{q}\,δ_{\mathbb{S}}\,\nabla u + |u|^{p-1}u \qquad \text{in } (0,\infty)\times\mathbb{R}^N, ] where $|\mathfrak{q}|\le 1$, $p>1$, and $\mathbb{S}$ is a fixed interface hyperplane. Working in Lebesgue spaces, we first establish local well-posedness of mild solutions. This is achieved by combining Gaussian bounds for the associated fundamental solution with a contraction mapping argument adapted to the lack of spatial homogeneity induced by the interface term. We then prove a sharp Fujita-type dichotomy for nonnegative solutions. Specifically, we show that every nontrivial solution blows up in finite time when $1<p \le 1+\frac{2}{N}$, whereas for $$p\>1+\frac{2}{N}$$ global solutions exist for sufficiently small initial data. The blow-up analysis relies on a suitably adapted test-function method that accounts for the presence of the interface. It is noteworthy that the critical exponent coincides with the classical Fujita exponent for the heat equation, indicating that the Fujita phenomenon remains stable under the presence of discontinuous diffusion effects and interface transmission conditions. To the best of our knowledge, this is the first result of this type for operators involving a singular drift supported on a hypersurface.

• The paper proves that the sharp Fujita critical exponent remains p_F=1+2/N, despite the interface operator’s lack of translation invariance and scale homogeneity.
• The analysis derives two-sided Gaussian kernel bounds, establishes local well-posedness in Lebesgue spaces, and constructs global small-data solutions when p>1+2/N.
• For 1<p≤1+2/N, every nonnegative weak solution with positive L¹ mass blows up in finite time because even test functions cancel the interface term exactly.

This paper studies the Cauchy problem for a semilinear parabolic equation driven by an interface operator with singular drift,

$$\partial_t u = \mathscr{L}u + |u|^{p-1}u, \qquad \mathscr{L} = \Delta + 2\mathfrak{q}\,\delta_S\,\nabla,$$

on $(0,\infty)\times\mathbb{R}^N$, $N\ge 2$, where $S=\{x_N=0\}$ is a fixed hyperplane and $|\mathfrak{q}|\le 1$. The operator acts as the Laplacian on each half-space but imposes the transmission condition $$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$, so that $\mathfrak{q}$ controls the imbalance of diffusive flux across $S$; $\mathfrak{q}=0$ recovers the classical heat equation. The main result is that the Fujita critical exponent for this problem is exactly $p_F=1+2/N$, identical to the classical value, despite the fact that $(0,\infty)\times\mathbb{R}^N$0 is neither translation invariant nor scale homogeneous. To the authors' knowledge this is the first Fujita-type result for operators with singular drift supported on a hypersurface.

Probabilistic and analytic structure of the interface operator

The paper builds on Portenko's theory of diffusions with generalized drifts supported on hypersurfaces and on the explicit fundamental solution of Mastrangelo–Talbi. The kernel admits a factorization into the tangential Gaussian heat kernel in $(0,\infty)\times\mathbb{R}^N$1 and a one-dimensional skew kernel

$(0,\infty)\times\mathbb{R}^N$2

which is the transition density of skew Brownian motion. The associated Feller process has tangential component equal to standard Brownian motion (independent of the interface) and normal component solving $(0,\infty)\times\mathbb{R}^N$3, i.e. skew Brownian motion with skewness $(0,\infty)\times\mathbb{R}^N$4.

The central analytic estimate is a two-sided Gaussian comparison:

$(0,\infty)\times\mathbb{R}^N$5

proved directly from the explicit formula using $(0,\infty)\times\mathbb{R}^N$6. This comparison implies that the interface affects only multiplicative constants while leaving the diffusive scaling intact — the structural reason the Fujita exponent survives. The paper also establishes mass conservation in the forward variable, an asymmetric normalization in the reverse variable involving $(0,\infty)\times\mathbb{R}^N$7 (reflecting non-self-adjointness), and $(0,\infty)\times\mathbb{R}^N$8 bounds of the same form as the heat kernel. A Chapman–Kolmogorov identity is proved by lifting the one-dimensional semigroup property of $(0,\infty)\times\mathbb{R}^N$9 through the tangential-normal factorization; the one-dimensional proof requires a delicate half-line reduction exploiting a reflection identity for the heat kernel. These properties yield a conservative, positive, strongly continuous contraction semigroup on $N\ge 2$0 and $N\ge 2$1, with the standard $N\ge 2$2–$N\ge 2$3 smoothing estimate.

The paper also proves the equivalence of three formulations: the weak PDE formulation (with the explicit interface term $N\ge 2$4), the martingale problem for test functions satisfying the skew transmission condition, and the skew Brownian SDE. The key algebraic step is an Itô–Tanaka computation showing that the local-time coefficient $N\ge 2$5 vanishes exactly when the transmission condition holds.

Local well-posedness in Lebesgue spaces

The local theory covers $N\ge 2$6 under the condition $N\ge 2$7, or $N\ge 2$8 with $N\ge 2$9. Because the kernel is not a convolution kernel, the standard translation-based fixed-point argument fails; instead the contraction is built from pointwise Gaussian bounds yielding heat-type smoothing estimates. In the supercritical regime the fixed point lives in a weighted space with norm $S=\{x_N=0\}$0, $S=\{x_N=0\}$1; continuous dependence follows via a linearized equation with coefficient $S=\{x_N=0\}$2 bounded in $S=\{x_N=0\}$3, combined with a singular Gronwall lemma and, in the sub-case $S=\{x_N=0\}$4, a time-splitting argument applying the linear regularity lemma on shifted intervals. Positivity preservation is obtained from Picard iterates and approximation.

At the critical endpoint $S=\{x_N=0\}$5, the supercritical construction breaks because $S=\{x_N=0\}$6 makes the time integral divergent. The authors introduce an auxiliary exponent $S=\{x_N=0\}$7 and exploit smallness of the linear flow in an initial layer, quantified by a compact-set version of the vanishing estimate $S=\{x_N=0\}$8. Notably, in this critical case the constant in the continuous dependence estimate is independent of $S=\{x_N=0\}$9. Existence times are uniform on compact subsets of $|\mathfrak{q}|\le 1$0 in both regimes.

The doubly critical case $|\mathfrak{q}|\le 1$1 is explicitly left open: the authors remark that for certain $|\mathfrak{q}|\le 1$2 even local solutions may fail to exist, paralleling known pathologies for the classical semilinear heat equation with singular initial data.

Global existence for small data

For $|\mathfrak{q}|\le 1$3 and $|\mathfrak{q}|\le 1$4 with $|\mathfrak{q}|\le 1$5 sufficiently small, global mild solutions are constructed in the space $|\mathfrak{q}|\le 1$6, where $|\mathfrak{q}|\le 1$7 is chosen so that $|\mathfrak{q}|\le 1$8 and the beta-function integrals converge. The argument is a standard contraction in $|\mathfrak{q}|\le 1$9, relying only on the heat-type smoothing estimates; the interface enters solely through constants. Together with the blow-up result below, this yields a complete dichotomy at $$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$0.

Blow-up below the Fujita exponent

The blow-up theorem states that if $$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$1 and $$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$2 with $$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$3, every nonnegative weak solution blows up in finite time. The proof is by contradiction via the test-function method, with the decisive observation that the radial test function

$$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$4

is even in the normal variable, hence $$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$5 identically, so the interface term in the weak formulation vanishes exactly. The weak formulation therefore reduces to the classical identity, and Young's inequality produces the functional inequality

$$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$6

In the subcritical regime the right-hand side decays as $$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$7, contradicting $$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$8. At the critical exponent the bound is uniform in $$(1-\mathfrak{q})\partial_{x_N}f(\tilde x,0^+)=(1+\mathfrak{q})\partial_{x_N}f(\tilde x,0^-)$$9, giving $\mathfrak{q}$0; a refined estimate retaining the $\mathfrak{q}$1-dependence ($\mathfrak{q}$2, $\mathfrak{q}$3) then lets one pass first $\mathfrak{q}$4 (using vanishing of the tail integral) and then $\mathfrak{q}$5, again contradicting $\mathfrak{q}$6. The argument is entirely analytic, depending only on positivity and two-sided Gaussian bounds, and thus extends to any interface operator admitting comparable kernel estimates.

Limitations and open questions

Several restrictions are acknowledged. The geometry is limited to a flat hyperplane: replacing $\mathfrak{q}$7 by a smooth compact hypersurface destroys the tangential–normal factorization underlying all estimates, and whether curvature shifts the threshold remains open. The skewness parameter is constant; for $\mathfrak{q}$8-dependent or state-dependent $\mathfrak{q}$9 it is unclear whether the two-sided Gaussian comparison holds uniformly, particularly when $S$0. Extensions to fractional diffusion $S$1 with an analogous interface drift, to multiple parallel interfaces or networks (where multiple reflections complicate the kernel estimates), and to moving or random interfaces — for which even well-posedness appears open — are all posed as unresolved problems. The doubly critical Lebesgue case noted above is likewise open.

Conclusion

The paper establishes local well-posedness of mild solutions in Lebesgue spaces and a sharp Fujita dichotomy for a semilinear heat equation whose diffusion carries a singular drift concentrated on a hyperplane. The critical exponent $S$2 persists because the interface modifies only the constants in the Gaussian kernel bounds, not the scaling exponents; technically, the blow-up proof hinges on the exact cancellation of the interface term for even-in-$S$3 test functions. The results demonstrate stability of the Fujita phenomenon under discontinuous transmission conditions and provide a template — two-sided Gaussian comparison plus adapted test functions — applicable to other interface operators with comparable kernels.