Browsing by Author "Cortazar, Carmen"

Now showing 1 - 17 of 17
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    A nonlocal diffusion problem with a sharp free boundary
    (2019) Cortazar, Carmen; Quiros, Fernando; Wolanski, Noemi
    We introduce and analyze a nonlocal free boundary problem which may be of interest to describe the spreading of populations in hostile environments. The rate of growth of the volume of the region occupied by the population is proportional to the rate at which the total population decreases. We prove existence and uniqueness for the problem posed on the line, on the half-line with constant Dirichlet data, and in the radial case in several dimensions. We also describe the asymptotic behaviour of both the solution and its free boundary.
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    A semilinear problem associated to the space-time fractional heat equation in RN
    (2024) Cortazar, Carmen; Quiros, Fernando; Wolanski, Noemi
    We study the fully nonlocal semilinear equation partial derivative(alpha )(t)u +(-Delta)(beta )u = |u|(p-1 )u, p >= 1, where partial derivative t alpha stands for the usual time derivative when alpha=1 and for the Caputo alpha-derivative if alpha is an element of (0, 1), while (-Delta)(beta), beta is an element of (0, 1], is the usual beta power of the Laplacian. We prescribe an initial datum in L-q(R-N). We give conditions ensuring the existence and uniqueness of a solution living in L-q(R-N) up to a maximal existence time T that may be finite or infinite. If T is finite, the L-q norm of the solution becomes unbounded as time approaches T, and u is said to blow up in L-q. Otherwise, the solution is global in time. For the case of nonnegative and nontrivial solutions, we give conditions on the initial datum that ensure either blow-up or global existence. Our weakest condition for global existence and our condition for blow-up are both related to the size of the averages of the initial datum in balls. As a corollary, every nonnegative nontrivial solution in L-q blows up in finite time if 1 < p < p(f ):= 1 + 2 beta/N whereas if p > p(f) there are both solutions that blow up and global ones. Noteworthy, the critical Fujita-type exponent pf does not depend on alpha. However, there is an important difference in the behavior of solutions in the critical case p = p(f) depending on the value of this parameter: when alpha = 1 it was known that all nonnegative and nontrivial solutions blow up, while we prove here that if alpha is an element of (0, 1) there is global existence for some initial data.
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    An inhomogeneous nonlocal diffusion problem with unbounded steps
    (SPRINGER BASEL AG, 2016) Cortazar, Carmen; Elgueta, Manuel; Garcia Melian, Jorge; Martinez, Salome
    We consider the following nonlocal equation
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    Asymptotic Behavior for a Nonlocal Diffusion Equation in Domains with Holes
    (2012) Cortazar, Carmen; Elgueta, Manuel; Quiros, Fernando; Wolanski, Noemi
    The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, u (t) = J*u-u := Lu, in an exterior domain, Omega, which excludes one or several holes, and with zero Dirichlet data on . When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves like a function that is L-harmonic, Lu = 0, in the exterior domain and vanishes in its complement. The height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior. The inner and the outer behaviors can be presented in a unified way through a suitable global approximation.
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    Asymptotic profiles for inhomogeneous heat equations with memory
    (2024) Cortazar, Carmen; Quiros, Fernando; Wolanski, Noemi
    We study the large-time behavior in all L-p norms of solutions to an inhomogeneous nonlocal heat equation in R-n involving a Caputo alpha-time derivative and a power beta of the Laplacian when the dimension is large, N > 4 beta. The asymptotic profiles depend strongly on the space-time scale and on the time behavior of the spatial L-1 norm of the forcing term.
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    Boundary fluxes for nonlocal diffusion
    (ACADEMIC PRESS INC ELSEVIER SCIENCE, 2007) Cortazar, Carmen; Elgueta, Manuel; Rossi, Julio D.; Wolanski, Noemi
    We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition. (c) 2006 Elsevier Inc. All rights reserved.
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    Decay/growth rates for inhomogeneous heat equations with memory. The case of large dimensions
    (2022) Cortazar, Carmen; Quiros, Fernando; Wolanski, Noemi
    We study the decay/growth rates in all L-p norms of solutions to an inhomogeneous nonlocal heat equation in R-N involving a Caputo alpha-time derivative and a power beta of the Laplacian when the dimension is large, N > 4 beta. Rates depend strongly on the space-time scale and on the time behavior of the spatial L-1 norm of the forcing term.
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    EXISTENCE AND ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SOME INHOMOGENEOUS NONLOCAL DIFFUSION PROBLEMS
    (SIAM PUBLICATIONS, 2009) Cortazar, Carmen; Elgueta, Manuel; Garcia Melian, Jorge; Martinez, Salome
    We consider the nonlocal evolution Dirichlet problem u(t)(x, t) = f(Omega) J(x-y/g(y)) u(y, t)/g(y)(N) dy- u(x, t), x is an element of Omega, t > 0; u = 0, x is an element of R-N\Omega, t >= 0; u(x, 0) = u(0)(x), x is an element of R-N; where Omega is a bounded domain in R-N, J is a Holder continuous, nonnegative, compactly supported function with unit integral and g is an element of C((Omega) over bar) is assumed to be positive in Omega. We discuss existence, uniqueness, and asymptotic behavior of solutions as t -> |infinity. Moreover, we prove the existence of a positive stationary solution when the inequality g(x) <= delta(x) holds at every point of Omega, where delta(x) = dist(x, partial derivative Omega). The behavior of positive stationary solutions near the boundary is also analyzed.
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    How to approximate the heat equation with neumann boundary conditions by nonlocal diffusion problems
    (2008) Cortazar, Carmen; Elgueta, Manuel; Rossi, Julio D.; Wolanski, Noemi
    We present a model for nonlocal diffusion with Neumann boundary conditions in a bounded smooth domain prescribing the flux through the boundary. We study the limit of this family of nonlocal diffusion operators when a rescaling parameter related to the kernel of the nonlocal operator goes to zero. We prove that the solutions of this family of problems converge to a solution of the heat equation with Neumann boundary conditions.
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    MULTIPLICITY RESULTS FOR GROUND STATE SOLUTIONS OF A SEMILINEAR EQUATION VIA ABRUPT CHANGES IN MAGNITUDE OF THE NONLINEARITY
    (2022) Cortazar, Carmen; Garcia-Huidobro, Marta; Herreros, Pilar
    Given k is an element of N, we define a class of continuous piecewise functions f having abrupt but controlled magnitude changes so that the problem Delta u + f (u) = 0, x is an element of RN, N > 2, has at least k radially symmetric positive solutions.
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    Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions
    (2009) Cortazar, Carmen; Elgueta, Manuel; Rossi, Julio D.
    We present a model for nonlocal diffusion with Dirichlet boundary conditions in a bounded smooth domain. We prove that solutions of properly rescaled nonlocal problems approximate uniformly the solution of the corresponding Dirichlet problem for the classical heat equation.
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    On the uniqueness of ground state solutions of a semilinear equation containing a weighted Laplacian
    (2006) Cortazar, Carmen; Garcia-Huidobro, Marta
    We consider the problem of uniqueness of radial ground state solutions to
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    On the uniqueness of sign changing bound state solutions of a semilinear equation
    (GAUTHIER-VILLARS/EDITIONS ELSEVIER, 2011) Cortazar, Carmen; Garcia Huidobro, Marta; Yarur, Cecilia S.
    We establish the uniqueness of the higher radial bound state solutions of
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    ON THE UNIQUENESS OF SOLUTIONS OF A SEMILINEAR EQUATION IN AN ANNULUS
    (2021) Cortazar, Carmen; Garcia-Huidobro, Marta; Herreros, Pilar; Tanaka, Satoshi
    We establish the uniqueness of positive radial solutions of
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    On the uniqueness of the second bound state solution of a semilinear equation
    (ELSEVIER SCIENCE BV, 2009) Cortazar, Carmen; Garcia Huidobro, Marta; Yarur, Cecilia S.
    We establish the uniqueness of the second radial bound state solution of
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    Stationary Sign Changing Solutions for an Inhomogeneous Nonlocal Problem
    (INDIANA UNIV MATH JOURNAL, 2011) Cortazar, Carmen; Elgueta, Manuel; Garcia Melian, Jorge; Martinez, Salome
    We consider the following nonlocal equation:
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    The blow-up problem for a semilinear parabolic equation with a potential
    (ACADEMIC PRESS INC ELSEVIER SCIENCE, 2007) Cortazar, Carmen; Elgueta, Manuel; Rossi, Julio D.
    Let Omega be a bounded smooth domain in R-N. We consider the problem u(t) = Delta u + V(x)u(P) in Omega x [0, T), with Dirichlet boundary conditions u = 0 on partial derivative Omega x [0, T) and initial datum u (x, 0) = M phi (x) where M >= 0, phi is positive and compatible with the boundary condition. We give estimates for the blow-up time of solutions for large values of M. As a consequence of these estimates we find that, for M large, the blow-up set concentrates near the points where phi(P-1) V attains its maximum. (c) 2007 Elsevier Inc. All rights reserved.