## Advances in Differential Equations

- Adv. Differential Equations
- Volume 12, Number 6 (2007), 669-720.

### Sturmian nodal set analysis for higher-order parabolic equations and applications

#### Abstract

We describe the local pointwise structure of multiple zeros of solutions of $2m$th-order
linear uniformly parabolic equations, \begin{equation} \tag*{(0.1)} u_t = \sum _{|{\beta}
| \le 2m} \, a_{\beta}(x,t) D^{\beta}_x u \quad {\rm in} \,\,\, {{\bf R}^N} \times [-1,1]
\quad (m \ge 2), \end{equation} with bounded and Lipschitz-continuous (for $|{\beta}|=2m$)
coefficients, in the existence-uniqueness class $\{|u(x,t)| \le B {\mathrm
e}^{b|x|^{\alpha}}\}$, 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}} \quad (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 there exists a countable family of multiple zeros evolving as $t
\to 0^-$ according to the nodal sets of *polynomial* eigenfunctions of a
non-self-adjoint operator ${{\mathbf B}}^*$ associated with that in (0.1). 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}} \quad (t>0). $ For the 1D second-order ($m=1$) parabolic equation with
smooth coefficients, $$ u_t= a(x,t) u_{xx} + q(x,t) u \quad (a(x,t) \ge a_0>0), $$ this
two-step analysis is known as {Sturm's Second Theorem} on zero sets, established by
C.~Sturm in 1836, [32]. 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 Poincaré--Bendixson theorems, mean curvature and curve
shortening flows, symplectic geometry, etc. Using such a local classification of multiple
zeros, we establish a unique continuation theorem for higher-order parabolic PDEs and
inequalities, and estimate the Hausdorff dimension of nodal sets of solutions. It turns
out that some common features of multiple zeros formation can be observed for solutions of
other PDEs including linear dispersion and wave equations, $$ u_t=u_{xxx} \quad \mbox{and}
\quad u_{tt}=u_{xx}, $$ and we present a discussion on this issue.

#### Article information

**Source**

Adv. Differential Equations, Volume 12, Number 6 (2007), 669-720.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867449

**Mathematical Reviews number (MathSciNet)**

MR2319452

**Zentralblatt MATH identifier**

1160.35010

**Subjects**

Primary: 35K30: Initial value problems for higher-order parabolic equations

Secondary: 35K45: Initial value problems for second-order parabolic systems

#### Citation

Galaktionov, V. A. Sturmian nodal set analysis for higher-order parabolic equations and applications. Adv. Differential Equations 12 (2007), no. 6, 669--720. https://projecteuclid.org/euclid.ade/1355867449