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We present baseline solutions in Sect.

2, we similarly analyze the timescale-interaction mechanisms that support SB solutions (in Sect. 3) and SS solutions (in Sect. 4).

We describe the problems in Sect. 3 and present our solutions in Sect. 4.

We follow similar arguments to those presented for (L^p({{mathbb R}^{N}}))-valued solutions in Sect.

We use it to set up a diffusion approximation with a noise term accounting for finite-size effects on traveling wave solutions in Sect. 4.

The SS system features bidirectional coupling between v and (ca_{i}), which is similar to that in the Jasinski model; therefore, we may also be able to explain the mechanisms underlying SS dynamics from the perspective of how the ((v, h)) system is driven by the ((ca_{i}, c_{mathrm{tot}}, l)) system in analogy to our approach for SB solutions in Sect.

We follow a similar analysis to the previous case, but identify the differences in the optimal control structure from the solution in Sect.

Then, the influences of crystalline anisotropy and indenter size on hardness, contact stress distribution, critical load for first dislocation emission and strain energy under the indenter are discussed, and the simulation results are compared with experimental results and Rice-Thomson (R-T) dislocation model solution in Sect.

As in Fig.  7a, the pressure distributions in the innermost lobules in Fig.  7b are similar to that in the single lobule solution in Sect.

These are the complex geometrical optics (CGO) solutions considered in Sect.

4, we prove interior and boundary (L_infty ) and Hölder estimates for derivatives of solutions, while in Sect. 5 we establish the interior and boundary mean oscillation estimates for the system in the whole space and in a half space.