Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions

By Paul C. Fife. 2003

379 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/2a9d7d89ca41f2750025a532a31b40920d174c29

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Citations in the corpus

Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations (2014, background): “2) are often used to model diffusive systems which exhibit nonlocal internal interactions (see [4, 7, 10, 11, 12, 16, 19, 24, 26], etc.”
The phase field method for geometric moving interfaces and their numerical approximations (2019, neutral): “A number of studies on nonlocal Allen-Cahn and nonlocal Cahn-Hilliard can be found in [51, 48, 55, 109, 108, 148, 214, 275].”
On Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications (2013, background): “Nonlocal dispersal operator such as u(x) 7→ ∫ RN k(y−x)[u(y)−u(x)]dy is applied when diffusive systems exhibit long range internal interactions (see [15], [17], [21]).”
The Neumann problem for nonlocal nonlinear diffusion equations (2008, methodology): “As stated in [ 16 ], if u(t;x) is thought of as the density of a single population at the point x at time t, and J(xiy) is thought of as the probability distribution of jumping from location y to location x, then the convolution (J ⁄ u)(t;x) = R RN J(y i x)u(t;y)dy is the rate at which individuals a”
Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints (2012, background): “6], kinetic equations [12, 38], phase transitions [15, 31], nonlocal heat conduction [16], and the peridynamic model for mechanics [48] and its one-dimensional variants [49, 50] for which L arises directly.”
Boundary fluxes for nonlocal diffusion (2007, background): “As stated in [7] if u(x, t) is thought of as a density at the point x at time t and J (x − y) is thought of as the probability distribution of jumping from location y to location x, then ∫ RN J (y−x)u(y, t) dy = (J ∗u)(x, t) is the rate at which individuals are arriving at position x from all other ”
Decay estimates for nonlocal problems via energy methods (2009, background): “This equation shares many properties with the classical heat equation, ut = u, such as: bounded stationary solutions are constant, a maximum principle holds for both of them and, even if G is compactly supported, perturbations propagate with infinite speed [20].”
A nonlocal convection–diffusion equation (2007, background): “As we have mentioned in the introduction, when J is nonnegative and compactly supported, this equation shares many properties with the classical heat equation, wt = ∆w, such as: bounded stationary solutions are constant, a maximum principle holds for both of them and perturbations propagate with inf”
Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems (2013, methodology): “Among those applications we have image analyses [14, 24, 25, 32], machine learning [36], nonlocal Dirichlet forms [6], kinetic equations [8, 31], phase transitions [11, 23], and nonlocal heat conduction [12].”