Neumann boundary conditions for a nonlocal nonlinear diffusion operator. Continuous and discrete models
Let J: ℝ → ℝ be a nonnegative, smooth function with ∫ℝ J(r)dr = 1, supported in [-1, 1], symmetric, J(r) = J(-r), and strictly increasing in [-1,0]. We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation ut(x,...
| Authors: | ; ; |
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| Format: | Article |
| Status: | Published version |
| Publication date: | 2007 |
| Country: | Argentina |
| Institution: | UBA-FCEN |
| Repository: | Biblioteca Digital (UBA-FCEN) |
| Language: | English |
| OAI Identifier: | paperaa:paper_00029939_v135_n12_p3837_Bogoya |
| Online Access: | http://hdl.handle.net/20.500.12110/paper_00029939_v135_n12_p3837_Bogoya |
| Access Level: | Open access |
| Keyword: | Neumann boundary conditions Nonlocal diffusion |
| Summary: | Let J: ℝ → ℝ be a nonnegative, smooth function with ∫ℝ J(r)dr = 1, supported in [-1, 1], symmetric, J(r) = J(-r), and strictly increasing in [-1,0]. We consider the Neumann boundary value problem for a nonlocal, nonlinear operator that is similar to the porous medium, and we study the equation ut(x, t)=∫L-L(J(x-y/ u(y,t) - J(x-y/u(x, t))dy, x∈[-L, L].We prove existence and uniqueness of solutions and a comparison principle. We find the asymptotic behaviour of the solutions as t → ∞: they converge to the mean value of the initial data. Next, we consider a discrete version of the above problem. Under suitable hypotheses we prove that the discrete model has properties analogous to the continuous one. Moreover, solutions of the discrete problem converge to the continuous ones when the mesh parameter goes to zero. Finally, we perform some numerical experiments. © 2007 American Mathematical Society. |
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