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Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions

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  • We study the qualitative behavior of a system of parabolic conservation laws, derived from a Keller-Segel type chemotaxis model with singular sensitivity, on the unit square or cube subject to various types of boundary conditions. It is shown that for given initial data in $H^3(\Omega)$, under the assumption that only the entropic energy associated with the initial data is small, there exist global-in-time classical solutions to the initial-boundary value problems of the model subject to the Neumann-Stress-free and Dirichlet-Stress-free type boundary conditions; these solutions converge to equilibrium states, determined from initial and/or boundary data, exponentially rapidly as time goes to infinity. In addition, it is shown that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model as the chemical diffusion rate tends to zero under the Neumann-Stress-free type boundary conditions. Numerical analysis is performed for a discretization of the model with the Dirichlet-Stress-free type boundary conditions, and a monotonic exponential decay to the equilibrium solution (analogous to the continuous case) is proven. Numerical simulations are supplemented to illustrate the exponential decay, test the assumptions of the exponential decay theorem, and to predict boundary layer formation under the Dirichlet-Stress-free type boundary conditions.

    Mathematics Subject Classification: 35A01, 35B40, 35K61, 35L65, 35Q92, 92C17.

    Citation:

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  • Figure 1.  Shown above is decay in time of solution norms to Algorithm 5.1, with varying $ \varepsilon $

    Figure 2.  Shown above are plots of solutions for $ \varepsilon = 0.001 $ and varying times, for Numerical Test 3. Contours of $ p $ (top) and $ | {\bf q}| $ (middle row) are shown at various times, and the bottom row shows plots of $ p $ and $ {\bf q}_2 $ with fixed $ x = 0.5 $ and varying $ t $

    Figure 3.  Shown above is a plot of $ | {\bf q}| $ with fixed $ x = 0.5 $ and $ t = 0.2 $ (right), for varying $ \varepsilon $ in numerical experiment 3

    Figure 4.  Shown above are solution plots for $ \varepsilon = 0.01 $ at varying times, for numerical experiment 4. Across the top row are contour plots of solutions $ p $, in the middle row are contours for $ | {\bf q}| $, and in the bottom row we show plots of $ p $ (left) and $ {\bf q}_2 $ (right) along $ x = 0 $

    Figure 5.  Shown above are plots of $ (\| p(t)\|^2 + \| {\bf q}(t) \|^2) $ versus time $ t $ (left), and $ {\bf q}_2(0,y,0.1) $, for varying $ \varepsilon $, for numerical experiment 4

    Table 1.  Values of energy norms at the end time $ T = 1 $ for various $ P^* $ values

    Backward Euler
    $ P^* $ $ \left( \| p_h(T) \|^2 + \| {\bf q}_h(T) \|^2 \right)^{1/2} $
    50 433.1920
    51 913.7230
    52 5.3532e+03
    53 1.4410e+04
    54 2.7783e+04
    55 4.5390e+04
    56 7.4976e+04
    57 3.6508e+11
    58 1.2408e+13
     | Show Table
    DownLoad: CSV

    Table 2.  Values of the critical $ P^* $ for blowup, for varying time step sizes

    $ \Delta t $ $ P^*_{critical} $
    0.01 57
    0.005 57
    0.0025 64
    0.00125 90
    0.000625 91
    0.0003125 90
    0.00015625 90
     | Show Table
    DownLoad: CSV
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