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Sturmian nodal set analysis for higher-order parabolic equations and applications. (English) Zbl 1160.35010

Summary: We describe the local pointwise structure of multiple zeros of solutions of \(2m\)th-order linear uniformly parabolic equations
\[ u_t= \sum_{|\beta|\leq2m} a_\beta(x,t)D_x^\beta u\quad\text{in }\mathbb R^N\times[-1,1] \qquad (m\geq 2), \tag{1} \]
with bounded and Lipschitz-continuous (for \(|\beta|=2m\)) coefficients, in the existence-uniqueness class \(\{|u(x,t)|\leq Be^{b|x|^\alpha}\}\), where \(B,b>0\) are constants and \(\alpha= \frac{2m}{2m-1}\). Assuming that \(u(0,0)=0\) and using the Sturmian backward continuation blow-up variable \(y=x/(-t)^{\frac{1}{2m}}\) \((t<0)\), we perform a classification of all possible types of formation as \(t\to 0^-\) of multiple spatial zeros of the solutions \(u(x,t)\). We show that ther exists a countable family of multiple zeros evolving as \(t\to 0^-\) according to the nodal sets of polynomial eigenfunctions of non-selfadjoint operator \(\mathbb B^*\) associated with that in (1).
Next, we show that other related polynomial solutions occur in the collapse of multiple zeros as \(t\to 0^+\), which is described in terms of the forward continuation variable \(y=x/t^{\frac{1}{2m}}\) \((t>0)\).
For the 1D second-order \((m=1)\) parabolic equation with smooth coefficients
\[ u_t= a(x,t)u_{xx}+q(x,t)u \qquad (a(x,t)\geq a_0>0), \]
this two-step analysis is known as Sturm’s Second Theorem on zero sets, established by C. Sturm in 1836. His more famous First Theorem (the number of zeros of solutions is non-increasing with time) was derived as a consequence of the second one. In the last thirty years these PDE ideas of Sturm found new applications, generalizations and extensions in various areas of general parabolic theory, stability and orbital connection problems, unique continuation and PoincarĂ©-Bendixson theorems, mean curvature and curve shortening flows, symplectic geometry, etc.

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
35K30 Initial value problems for higher-order parabolic equations
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