What interior regularity do mixed local-nonlocal parabolic semilinear equations have?

The operator adds a fractional and a classical Laplacian

The equation studied is u_t = L u + f(u) on Ω × (0,T), with u(x,0) = u_0(x) on Ω and u = 0 on the whole complement R^n \ Ω for all times. The diffusion operator is L = -(-Δ)^s + Δ, where (-Δ)^s is the integral fractional Laplacian with s in (0,1) and Δ is the classical local Laplacian. The two terms are added, not compared through a small parameter, so the operator mixes a nonlocal mechanism with a local one and lacks scale invariance.

Because the fractional Laplacian is an integral over all of R^n, Dirichlet conditions must be imposed on the entire complement R^n \ Ω rather than only on the boundary ∂Ω. This is what makes the integral defining (-Δ)^s meaningful. The reaction term f is nonnegative, locally C^1 and increasing on its domain E = [0,α) for some α > 0, and the initial datum u_0 lies in H^1_0(Ω) ∩ L^∞(Ω) with u_0(x) in E almost everywhere. The paper notes that this restricted-domain reaction was inspired by a quenching problem with f(u) = λ(1-u)^{-p}, whose term blows up as the L^∞ norm of u approaches 1; the results here are stated in a simplified case that the authors say can be adapted to the general interval E = (α_1, α_2).

A unique classical solution exists up to a maximal time

Proposition 1.1 states that for every u_0 in H^1_0(Ω) ∩ L^∞(Ω) with u_0(x) in E almost everywhere, there is some T = T(u_0) > 0 such that the system admits a unique classical solution u in C((0,T), D(L)) ∩ C([0,T), H^1_0(Ω)) ∩ C^1((0,T), L^2(Ω)). The maximal existence time is characterized by the boundary of the reaction domain: limsup as t → t_0 of the L^∞(Ω) norm of u(·,t) equals α if and only if t_0 = T.

The proof first truncates f to a globally Lipschitz, increasing function f_ε that agrees with f on (0, α-ε), then treats the truncated problem. A bilinear form combining the fractional seminorm and the gradient integral has natural domain H^1_0(Ω) ∩ H^s_0(Ω) = H^1_0(Ω), and its associated norm is equivalent to the usual H^1_0 norm. By Kato's representation theorem this form yields a unique nonnegative self-adjoint operator A = -L = (-Δ)^s - Δ generating an analytic semigroup e^{Lt} in L^2(Ω). Classical semigroup theory then gives a unique classical solution of the truncated problem. Two comparison results follow: order preservation with respect to initial data, and a comparison principle between the solution and supersolutions or subsolutions. These let the authors keep the truncated solution inside (0, α-ε) up to some time T_ε, so it can be identified with a solution of the original problem.

Solutions are Hölder continuous in space and time

Theorem 1.2 fixes a point (x_0, t_0) in Ω × (0,T) and uses parabolic cylinders Q_ρ = B_ρ(x_0) × [t_0 - ρ^2, t_0]. For every α_0 in (0,1) there is some r* = r*(n, s, α_0, x_0, t_0) in (0,1) such that for every r ≤ r*, the averaged integral over Q_r of |u(x,t) - (u)_{Q_r}|^2 dx dt is bounded by c r^{2α_0}. The constant c depends on n, s, α_0, r*, the L^∞ norm of f(u) on Ω × [0,t_0], the L^2 norm of u on Q_{r*}, and the nonlocal tail Tail_∞(u, x_0, r*, t_0 - r*^2, t_0). In particular the solution is locally Hölder continuous of exponent α_0 in both space and time.

Two features of this statement matter for reading it correctly. First, the conclusion is local: it holds in an interior parabolic cylinder around a point with t_0 > 0, and the radius r* depends on that point. Second, the exponent α_0 is arbitrary in (0,1) but is chosen before r* is determined, so the estimate is not a single fixed Hölder exponent. The proof proceeds by a Campanato argument, which converts the decay of the averaged L^2 oscillation over shrinking parabolic cylinders into Hölder continuity. The nonlocal tail enters the constant because the fractional Laplacian sees values of u outside the ball B_r(x_0), so decay at infinity has to be controlled alongside the local oscillation.

An L2-to-L∞ bound controls the linear problem

Theorem 1.3 concerns the linear equation v_t = L v + a(x,t) v on Ω × (0,T'), with the same Dirichlet conditions on R^n \ Ω and initial datum v_0 in H^1_0(Ω), where a is in L^∞_loc(Ω × [0,T')). For (x_0, t_0) in Ω × (0,T'), τ in (0,1), R in (0,1) and Q_R = B_R(x_0) × (t_0 - R^2, t_0) with B_R(x_0) contained in Ω and [t_0 - R^2, t_0] contained in (0,T), the L^∞ norm of v on Q_{τR} is bounded by C R^{-n/2} τ^{-n/2-1} (1-τ)^{-(n/2+s)(n/2+1)} times the L^∞ norm of v over [t_0 - R^2, t_0] with values in L^2(Ω), where C depends on n, s and the L^∞ norm of a on Ω × [0,t_0].

This is a spatial L^2-to-L^∞ estimate: it upgrades an L^∞-in-time, L^2-in-space bound on a larger cylinder to a pointwise bound on a strictly smaller cylinder. The price is explicit in the factor (1-τ)^{-(n/2+s)(n/2+1)}, which blows up as the inner cylinder approaches the outer one, and in the negative powers of R and τ. The proof obtains a Caccioppoli inequality for the system and then applies a Moser iteration argument, using a lemma on sequences of positive numbers to drive the iteration to zero. The paper describes this estimate as a key ingredient for the final regularity result, and it is also applied to the difference quotients in time used later.

The time derivative is Hölder and space is C2,α

Theorem 1.4 states that if u is the unique classical solution of the semilinear system, then for every α_0 in (0,1) the time derivative u_t belongs to C^{α_0}_loc(Ω × [0,T)) and u(·,t) belongs to C^{2,α_0}_loc(Ω) for every t in (0,T). This is the paper's main objective: interior C^{2,α} regularity in space for the semilinear mixed local-nonlocal parabolic equation.

The proof introduces the difference quotient v_h(x,t) = (u(x,t+h) - u(x,t))/h. For fixed h > 0 it has the same regularity as u and converges strongly to u_t in C([t_1,t_2], L^2(Ω)) on compact time intervals. Its L^2(Ω) norm is bounded independently of h by the L^∞ norm of u_t on the relevant time interval. The quotient solves a linear equation of the form covered by Theorem 1.3 with coefficient a(x,t) = f'(ξ(x,t)), where ξ(x,t) lies between u(x,t+h) and u(x,t); in an interior cylinder ξ stays bounded away from 1 independently of h, so f'(ξ) is an L^∞_loc function uniformly in h. Theorem 1.3 and the uniform L^2 bound give a bound on v_h that does not depend on h, and repeating the Campanato reasoning yields the averaged oscillation estimate for v_h with a constant independent of h. Letting h → 0 transfers the estimate to u_t, so u_t is locally Hölder continuous for any exponent α_0 in (0,1).

The final step rewrites the equation as -Δu(x,t) + (-Δ)^s u(x,t) = f(u(x,t)) - u_t(x,t). For each fixed t in (0,T) the right-hand side is C^{α_0}_loc(Ω): f(u) inherits this from the Hölder continuity of u and the regularity of f, and u_t is Hölder by the previous step. Applying known elliptic regularity for the mixed operator L = -(-Δ)^s + Δ then gives u(·,t) in C^{2,α_0}_loc(Ω). The spatial regularity is therefore obtained by combining the new parabolic estimates with an elliptic regularity input rather than by a purely parabolic argument.

Interior regularity stops short of the boundary and t = 0

The regularity conclusions are interior in space and hold for positive times. Theorem 1.2 is stated around a point (x_0, t_0) in Ω × (0,T) with a radius r* depending on that point, and Theorem 1.4 gives u(·,t) in C^{2,α_0}_loc(Ω) for every t in (0,T), with u_t in C^{α_0}_loc(Ω × [0,T)). The results do not assert regularity up to the boundary ∂Ω or at t = 0, and the Dirichlet condition is imposed on the whole complement R^n \ Ω, so the boundary data are part of the nonlocal setting rather than a local trace condition.

The reaction term is also restricted: f is defined on E = [0,α), nonnegative, locally C^1 and increasing, and the maximal existence time T is tied to the L^∞ norm of u approaching α. The paper states that the results are formulated in a somewhat particular case that simplifies some arguments and that they could be adapted to the general case E = (α_1, α_2) with more work. The Hölder exponent α_0 is arbitrary in (0,1) but is selected before the radius r*, and the constants in Theorems 1.2 and 1.3 depend on the nonlocal tail, the L^∞ norm of f(u) or a, and the L^2 norm of the solution on a larger cylinder, so the estimates are not uniform in those quantities.

Hölder continuity and C2,α answer different questions

The paper reports two distinct levels of spatial regularity, and they should not be conflated. Theorem 1.2 gives local Hölder continuity of exponent α_0 in both space and time, which controls how much u can oscillate over small parabolic cylinders but says nothing about differentiability. Theorem 1.4 gives u(·,t) in C^{2,α_0}_loc(Ω) for every t in (0,T), which means second spatial derivatives exist and are themselves Hölder continuous of exponent α_0. The second statement is strictly stronger in space and is obtained only for positive times, using the Hölder continuity of u_t and elliptic regularity for the mixed operator.

The two results also rest on different machinery. The Hölder continuity comes from a Campanato argument on the semilinear problem, with the nonlocal tail entering the constant. The C^{2,α} result comes from the L^2-to-L^∞ estimate for the linear problem with a zeroth-order term, applied to the time difference quotient, followed by the elliptic regularity input. For a reader comparing this paper with work on purely local or purely fractional parabolic equations, the relevant distinction is that here the spatial smoothing to C^{2,α} is interior and for t > 0, while the time derivative is only Hölder continuous, not smoother.

The paper's priority is interior C2,α in space

The central claim to take from this paper is the combination of Proposition 1.1 and Theorem 1.4: for the semilinear mixed local-nonlocal parabolic equation with L = -(-Δ)^s + Δ and Dirichlet conditions on R^n \ Ω, a unique classical solution exists up to a maximal time T characterized by the L^∞ norm of u reaching α, and for every positive time the solution is C^{2,α_0}_loc in space for every α_0 in (0,1), with u_t locally Hölder continuous. The spatial C^{2,α} statement is the paper's stated main objective, and it is reached by feeding the new parabolic Hölder estimates and the L^2-to-L^∞ bound into known elliptic regularity for the mixed operator.

The practical reading is therefore about which regularity level applies where. If the question is whether the solution is continuous in space and time, Theorem 1.2 answers it with a local Hölder estimate in parabolic cylinders. If the question is whether second spatial derivatives exist and are Hölder continuous, Theorem 1.4 answers it for t in (0,T) and only in the interior of Ω. The boundary and the initial time are outside both statements, and the reaction term is restricted to a domain [0,α) whose endpoint governs the maximal existence time. The paper's own framing places the result against a mostly elliptic literature on the mixed operator and a thinner parabolic one, where earlier work had established interior C^α regularity for the homogeneous parabolic problem; the contribution here is the semilinear setting and the higher interior spatial regularity.

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