Oscillations of solutions of initial value problems for parabolic equations (Q2758287)

This interesting paper deals with oscillations of solutions for the parabolic equations \(\partial u/\partial t -\triangle u +p(t)u=0\) in \(\mathbb{R}^n\times (0,\infty)\) under the initial condition \(u(x,0)=\varphi (x)\) for \(x\in \mathbb{R}^n\). Here \(\triangle \) is the Laplacian in \({\mathbb{R}}^n\), \(p\in C([0;\infty); (0,\infty))\) and \(p(t)\) is bounded from above. Two function spaces \(\mathcal E[0,\infty)\) and \(\Phi \) are introduced: \(\mathcal E[0,\infty)\) is defined to be the set of all continuous functions \(u(x,t)\) defined in \({\mathbb{R}}^n\times [0,\infty)\) satisfying \(|u(x,t)|\leq L e^{|x|^{\beta }}\), \(x\in {\mathbb{R}}^n\), \(t\geq 0\), \(L>0\), \(0<\beta <2\) (\(L\) and \(\beta \) may depend on \(u\)); \(\Phi \) is the set of all functions \(\varphi (x)\) in \({\mathbb{R}}^n\) such that \(\varphi \), \(\partial \varphi / \partial x_i\), \(\partial^2 \varphi / \partial x_i\partial x_j\) (\(i,j=1,2,\dots ,n\)) are all continuous and bounded in \({\mathbb{R}}^n\). Making use of these function spaces the authors assume \(\varphi \in \Phi \) and the classical solution of the considered problem \(u\in \mathcal E[0,\infty)\). It is shown that if \(\widehat\varphi _X(t)=\int_{\mathbb{R}}^{\text{n}}H(x,t)\varphi (x+X) dx\) is oscillatory at \(t=\infty \) for some \(X\in {\mathbb{R}}^n\), then \(u\) is oscillatory in \({\mathbb{R}}^n\times (0,\infty)\) (\(H\) is the Green function of the heat equation). Another result is that if there exists a sequence of zeros \(\{t_n\}\) of \(\widehat \varphi _X(t) \) such that \(\lim_{n_\to \infty }t_n=0\), then there exists a sequence of zeros \(\{(\tilde x_n,\tilde t_n)\}_{n=1}^{\infty }\subset {\mathbb{R}}^n\times (0,\infty)\) of the solution \(u\) such that \(\lim_{n\to \infty }\tilde t_n=0\). In addition, three examples illustrate the theory.