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Browsing by Author "Gavioli, Andrea"

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  • A class of singular first order differential equations with applications in reaction-diffusion
    Publication . Enguiça, Ricardo Roque; Gavioli, Andrea; Sanchez, Luis
    We study positive solutions y(u) for the first order differential equation y' = q(cy(1/p) - f(u)) where c > 0 is a parameter, p > 1 and q > 1 are conjugate numbers and f is a continuous function in [0, 1] such that f(0) = 0 = f(1). We shall be particularly concerned with positive solutions y(u) such that y(0) = 0 = y(1). Our motivation lies in the fact that this problem provides a model for the existence of travelling wave solutions for analogues of the FKPP equation in one space dimension, where diffusion is represented by the p-Laplacian operator. We obtain a theory of admissible velocities and some other features that generalize classical and recent results, established for p = 2.
    2013-01Journal article Metadata only Show more
  • Solutions of second-order and fourth-order ODEs on the half-line
    Publication . Enguiça, Ricardo Roque; Gavioli, Andrea; Sanchez, Luis
    We start by studying the existence of positive solutions for the differential equation u '' = a(x)u - g(u), with u ''(0) = u(+infinity) = 0, where a is a positive function, and g is a power or a bounded function. In other words, we are concerned with even positive homoclinics of the differential equation. The main motivation is to check that some well-known results concerning the existence of homoclinics for the autonomous case (where a is constant) are also true for the non-autonomous equation. This also motivates us to study the analogous fourth-order boundary value problem {u((4)) - cu '' + a(x)u = vertical bar u vertical bar(p-1)u u'(0) = u'''(0) = 0, u(+infinity) = u'(+infinity) = 0 for which we also find nontrivial (and, in some instances, positive) solutions.
    2010-11-01Journal article Open access Show more