Global well-posedness, ill-posedness and long time behavior of solutions of the surface electromigration equation

Abstract This paper is concerned with the Cauchy problem and the long time behavior of the two-dimensional surface electromigration (SEM) equation. Firstly, we prove that the SEM equation is locally well-posed in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> , and globally well-posed in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> without smallness assumption. Secondly, we prove that the SEM equation is ill-posed in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> in the sense that the solution map is not C 2 . Finally, we get a local energy decay result of the global solution in a growing unbounded in time region but not containing the soliton region. Our well-posedness results extend the latest findings in Linares et al (2021 Nonlinearity 34 5213–33), where the authors proved that the SEM equation is locally well-posed in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>s</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> . The key ingredient in this paper is to rewrite the linear and nonlinear part in the SEM equation into symmetric separate dispersive terms on the two space variables and some divergence form, respectively. After this invertible process, we use two bilinear estimates in the modified Fourier restriction space introduced in Kinoshita (2021 Ann. Inst. Henri Poincare C 38 451–505) to relax the local well-posedness index from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>s</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> .