Asymptotic behavior of solutions of a nonlinear degenerate chemotaxis model

Figure 1.  Unstructured triangular mesh for the space domain $ {\Omega} = {\left({0,1}\right)}\times{\left({0,1}\right)} $ with 14336 acute angle triangles

Figure 2.  Plot of $ h({\left\|{q}\right\|}^{2}) $ as a function of $ {\left\|{q}\right\|}^{2} $defined by equation (13). When the chemosensitivity strength $ \zeta $ increases beyond the critical value $ \zeta_{c} $, $ h({\left\|{q}\right\|}^{2}) $ becomes negative for a finite range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $ marked with rhombi

Figure 3.  To the top: Distribution of positive eigenvalues $ \lambda_{q}^{+} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $. To the bottom: Distribution of negative eigenvalues $ \lambda_{q}^{-} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $

Figure 4.  Initial condition of the function $ u{\left({ {\mathbf{x}},t}\right)} $ given by equation (25) with a small perturbation around zero. 2D view of the function $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and a 3D view of its magnitude (to the right)

Figure 5.  First row from left to right: Nonlinear evolution of the function $ u{\left({ {\mathbf{x}},t}\right)} $ at $ t = 2.5 $, $ t = 325 $, and $ t = 997.5 $. Second row from left to right: Evolution of the heterogeneous stationary solutions at the same moments as for the evolution of $ u{\left({ {\mathbf{x}},t}\right)} $

Figure 6.  Similarities of patterns between the nonlinear evolution $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and the heterogeneous state (to the right)

Figure 7.  Time evolution of the difference in $ {L^{2}} $ between $ u{\left({ {\mathbf{x}},t}\right)} $ and the heterogeneous solution

Figure 8.  Plot of $ h({\left\|{q}\right\|}^{2}) $ as a function of $ {\left\|{q}\right\|}^{2} $defined by equation (13). When the death rate $ \beta $ decreases below the critical value $ \beta_{c} $, $ h({\left\|{q}\right\|}^{2}) $ becomes negative for a finite range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $ marked with rhombi, and pattern formation can be expected

Figure 9.  To the top: Distribution of positive eigenvalues $ \lambda_{q}^{+} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $. To the bottom: Distribution of negative eigenvalues $ \lambda_{q}^{-} $ with respect to the range of unstable wave numbers $ {\left\|{q}\right\|}^{2} $

Figure 10.  Initial condition of the function $ u{\left({ {\mathbf{x}},t}\right)} $ given by equation (25) with a small perturbation around zero.2D view of the function $ u{\left({ {\mathbf{x}},t}\right)} $ (to the left) and a 3D view of its magnitude (to the right)

Figure 11.  First row from left to right. Nonlinear evolution of the function $ u{\left({ {\mathbf{x}},t}\right)} $ at $ t = 10 $, $ t = 70 $, and $ t = 750 $. Second row from left to right. Evolution of the heterogeneous stationary solutions at the same moments as for $ u{\left({ {\mathbf{x}},t}\right)} $

Figure 12.  Similarities of patterns between the nonlinear evolution $ u({\left({ {\mathbf{x}},t}\right)} $ (to the left) and the heterogeneous state (to the right)