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Bifurcation From Stability to Instability for a Free Boundary Problem Arising in a Tumor Model

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Abstract

We consider a time-dependent free boundary problem with radially symmetric initial data: σ t − Δσ + σ = 0 if and σ(r,0)=σ0(r) in {r < R(0)} where R(0) is given. This is a model for tumor growth, with nutrient concentration (or tumor cells density) σ(r,t) and proliferation rate then there exists a unique stationary solution (σ S (r), R S ), where R S depends only on the number . We prove that there exists a number μ *, such that if μ < μ * . . . then the stationary solution is stable with respect to non-radially symmetric perturbations, whereas if μ > μ * then the stationary solution is unstable.

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio, 43210, USA

    Avner Friedman

  2. Department of Mathematics, University of Notre Dame, 255 Hurley Hall, Notre Dame, Indiana, 46556

    Bei Hu

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  1. Avner Friedman
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  2. Bei Hu
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Correspondence to Avner Friedman.

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Communicated by P.H.Rabinowitz

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Friedman, A., Hu, B. Bifurcation From Stability to Instability for a Free Boundary Problem Arising in a Tumor Model. Arch. Rational Mech. Anal. 180, 293–330 (2006). https://doi.org/10.1007/s00205-005-0408-z

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  • DOI: https://doi.org/10.1007/s00205-005-0408-z

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