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How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. (English) Zbl 1145.35060

Let \(\Omega\) be a bounded, connected, smooth domain in \(\mathbb R^N\). We consider the mixed Cauchy-boundary value problem
\[ \begin{aligned} u_t - \Delta u = 0 &\quad\text{in } \Omega \times (0, T),\\ \frac{\partial u}{\partial \eta} = g &\quad\text{on } \partial \Omega \times (0, T),\\ u(x,0) = u_0(x) &\quad\text{in } \Omega. \end{aligned} \]
Let \(J: \mathbb R^N \to \mathbb R\) be a non-negative, radial, continuous function with \(\int_{\mathbb R^N} J(z)\, dz = 1\), strictly positive if \(|z| < d\) and vanishing if \(|z| \geq d\) and let \(G(x,\xi)\) be smooth and compactly supported in \(\xi\), uniformly in \(x\). If \(\varepsilon > 0\), we set \[ J_\varepsilon(\xi):= \frac{C_1}{\varepsilon^N} J(\xi/\varepsilon), G_\varepsilon(x,\xi):= \frac{C_1}{\varepsilon^N} G(x,\xi/\varepsilon), \]
with \(C_1^{-1}:= \frac{1}{2} \int_{B(0,d)} J(z) z_N^2\, dz\) and consider the initial value problem
\[ \begin{aligned} u^\varepsilon_t (x,t) &= \frac{1}{\varepsilon^2} \int_\Omega J_\varepsilon(x-y) (u^\varepsilon(y,t) - u^\varepsilon(x,t))\,dy+ \frac{1}{\varepsilon} \int_{\mathbb R^N \setminus \Omega} G_\varepsilon (x,x-y) g(y,t) \,dy,\\ u^\varepsilon_t (x,0) &= u_0(x). \end{aligned}\tag{2} \]
Then it is shown that, under suitable assumptions (2) is uniquely solvable. Moreover, some results of convergence of \(u^\varepsilon\) to \(u\) as \(\varepsilon \to 0\) are proved.

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
35A35 Theoretical approximation in context of PDEs
35K05 Heat equation
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References:

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