In this paper, we study the Cauchy problem for a heat equation governed by a mixed local--nonlocal diffusion operator with spatially irregular coefficients. We first establish classical well-posedness in an energy framework for bounded, measurable coefficients that satisfy uniform positivity, and we derive an a priori estimate ensuring uniqueness and continuous dependence on the initial data. We then extend the notion of solution to distributional coefficients and initial data by a Friedrichs-type regularisation procedure. Within this very weak framework, we establish the existence and uniqueness of solution nets and prove consistency with the classical weak solution whenever the coefficients are regular.
Motivated by ecological models in which individuals move by a superposition of (i) local Brownian diffusion and (ii) long-range Lévy-type jumps, Dipierro and Valdinoci [20] derived evolution equations whose dispersal mechanism is governed by the mixed operator L 0 u := -∆u + (-∆) s u, s ∈ (0, 1).
(1.1)
Here the local part -∆ describes short-range random motion, whereas the fractional Laplacian (-∆) s encodes long-range interactions and jump processes. In the whole space R d , (-∆) s may be defined via the principal value integral (-∆) s u(x) = c d,s P.V.
|x -y| d+2s dy, so that L 0 corresponds (up to sign conventions) to the infinitesimal generator of a stochastic motion obtained by superposing a Brownian component with an independent symmetric α-stable jump process, α = 2s [11,12]. In bounded domains, the mixed nature of (1.1) leads naturally to boundary conditions that combine a classical Neumann constraint for the local part with a genuinely nonlocal flux condition for the fractional component, as proposed in [20]. A systematic PDE analysis of elliptic problems driven by (1.1) was later developed in [6].
The mixed local-nonlocal operator L 0 has attracted considerable attention in recent years and has been investigated from several perspectives. This sustained interest is largely driven by its ability to capture mechanisms in which local diffusion coexists with long-range interactions and jump-like dispersal. In probability theory, operators of the form -∆ + (-∆) s arise as infinitesimal generators of stochastic dynamics obtained by superimposing a Brownian motion with an independent Lévy jump process, thereby modelling trajectories with frequent small displacements interspersed with occasional long relocations. In mathematical biology and ecology, the same superposition naturally appears in models of animal movement and optimal foraging, where long-range relocation (often idealised via Lévy flights) complements local exploratory behaviour and leads to mixed dispersal laws. Similar hybrid effects are also used to describe anomalous transport in complex environments (e.g. dispersal in porous or turbulent media), heterogeneous materials in which short-range diffusion is coupled with nonlocal transfer, and diffusion with trapping, absorption, or killing, where lower-order potential terms represent loss mechanisms or reactions. For further background and additional references, we refer the reader to [13,17,18,19,7,24,33,34] and the works cited therein.
In this paper, we extend (1.1) by allowing variable coefficients and consider the Cauchy problem
where L is the mixed local-nonlocal operator
Here u denotes a scalar state variable (e.g., temperature, concentration, density), u 0 is the initial datum. The coefficients a, b, c are assumed real-valued, with a(x) ≥ a 0 > 0, b(x) ≥ b 0 > 0, and c(x) ≥ c 0 ≥ 0 for a.e. x ∈ R d . Moreover, (-∆) s/2 denotes the (Riesz) fractional Laplacian. Note that when b ≡ const, the fractional term reduces to b (-∆) s u.
More precisely, the divergence-form term -∇• (a(x)∇u) models heterogeneous classical diffusion, whereas the fractional component (-∆) s/2 b(x)(-∆) s/2 u incorporates nonlocal effects of order 2s modulated by b(x). The lower-order term c(x)u may be interpreted, depending on the application, as reaction, damping, or absorption.
We assume that the coefficients u 0 , a, b, c ∈ D ′ (R) are irregular, distributional in space. Once the coefficients are distributional, one encounters the classical obstruction pointed out by Schwartz: in general, there is no consistent product of distributions extending the pointwise product of smooth functions [31]. Consequently, the equation cannot be interpreted in a classical sense, and classical (or even standard weak) solution frameworks typically break down when distributional coefficients must be multiplied.
To overcome this difficulty, we employ the notion of very weak solutions, introduced by Garetto and the second author [22]. This approach is based on regularising the distributional coefficients and data, solving a family of smooth problems, and identifying a solution concept through suitable moderateness and consistency properties.
This approach has proved effective for second-order hyperbolic and parabolic equations with non-regular coefficients [28,29], including equations with singular time-dependent potentials and other distributional terms; in and further extended in [1,2,3,4,14,32,15,16,8,30,26,9,10,25,23,5].
In the present work, we extend this methodology to the heat equation driven by mixed local-nonlocal operator (1.2) with variable coefficients a, b, c ∈ D ′ (R) modelled by the space of distributions.
The novelty of this work lies in the rigorous analysis of a parabolic equation with a mixed local-nonlocal diffusion operator and spatially irregular coefficients, a setting that has not been previously studied in a unified analytical framework. The results provide a foundation for further analytical and n
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