##
**Superlinear parabolic problems. Blow-up, global existence and steady states.
2nd revised and updated edition.**
*(English)*
Zbl 1423.35004

Birkhäuser Advanced Texts. Basler Lehrbücher. Cham: Birkhäuser (ISBN 978-3-030-18220-5/hbk; 978-3-030-18222-9/ebook). xvi, 725 p. (2019).

This book is devoted to the qualitative study of solutions of superlinear elliptic and parabolic partial differential equations and systems. As the authors say in the introduction, superlinear means that the problems involve nondisipative terms which grow faster than linearly for large values of the solutions. The book is structured in five main chapters and contains also a part of preliminaries, a bibliography, a list of symbols and an index. First chapter deals with elliptic problems. Here, the authors, study the following problem: \begin{align*} -\Delta u=f(x,u),\quad & x\in\Omega\\ u=0,\quad & x\in\partial\Omega \end{align*} where \(f:\Omega\times \mathbb{R}\to \mathbb{R}\) is a Caratheodory function. The mainly case considered is when \(f(x,u)=|u|^{p-1}u+\lambda u\), \(p>1\),\(\lambda\in \mathbb{R}\). The chapter contains 13 sections (1–13) and the topics covered are classical and weak solutions, singularities and a proiri bounds via different methods like Hardy-Sobolev inequalities, bootstrap in \(L_{\delta}^{p}\)-spaces, rescaling or moving planes. The second chapter is devoted to parabolic problems. The authors deal mainly with semilinear parabolic problems of the form \begin{align*} u_{t}-\Delta u=f(u),\quad & x\in\Omega,\quad t>0\\ u=0,\quad & x\in\partial\Omega,\quad t>0\\ u(x,0)=u_0(x),\quad & x\in\Omega \end{align*} where f is \(C^1\)-function with a superlinear growth. Most of the assertions are formulated for the case \(f(u)=|u|^{p-1}u\), \(p>1\). The 16 sections of the chapter (14–29), cover topics like well posedness, global existence, blow-up rate or a priori bounds.

Chapter 3 is devoted to elliptic and parabolic systems. In the case of elliptic systems (Section 31), the authors study the problem of a priori estimates and existence for weakly coupled systems. Section 32 deals with parabolic systems coupled by power source terms. Problems like well posedness, global existence and blow-up rate are studied here. In the last section (33), the authors study the different possible effects of adding linear diffusion to a system of ordinary differential equations. The results lead to the idea to consider some systems arising in biological or physical contexts such as mass-preserving and Grierer-Meinhardt systems. Chapter 4 is devoted to equations with gradient terms. The authors consider problems with nonlinearities depending on u and its space derivatives: \begin{align*} u_{t}-\Delta u=F(u,\nabla u),\quad & x\in\Omega,\quad t>0 u=0,\quad & x\in\partial\Omega,\quad t>0 u(x,0)=u_0(x),\quad & x\in\Omega \end{align*} where \(F=F(u,\xi):\mathbb{R}\times \mathbb{R}^n\to \mathbb{R}\) is \(C^1\)-function. In the Sections 36 to 39, \(F(u,\nabla u)=u^{p}+g(u,\nabla u)\) and only the nonnegative solutions are considered. In the Sections 40 and 41, the authors consider problems whose essential superlinear character comes from the gradient term. In these sections \(F(u,\nabla u)=|\nabla u|^{p}\), \(p>1\) in which case the problem arises in stochastic control theory, being also related with Kadar-Parisi-Zhang equation in the physical theory of growth and roughening of surfaces. The last chapter is devoted to various problems with nonlocal nonlinearities. The equations that are considered involve nonlocal terms taking the form of an integral in space or in time. In sections 43 and 44, the authors consider several equations with space integrals like: \[ u_{t}-\Delta u=\int_{\Omega}u^{p}(y,t)dy-ku^{q},\quad x\in\Omega,\quad t>0, \] where \(\Omega\subset \mathbb{R}^n\) is a bounded domain, \(p>1\), \(q\geq 1\), \(k\geq 0\), \(u_0\in L^{\infty}(\Omega)\), \(u_0\geq 0\), or \[ u_{t}-\Delta u=(\int_{\Omega}g(u)dx)^{m}f(u),\quad x\in\Omega,\quad t>0,\quad m\in \mathbb{R}^{\ast}. \] The global existence, blow-up and a priori estimates are studied. In Section 45, the authors deal with Cauchy problems with non local source terms involving space integrals of the form \begin{align*} u_{t}-\Delta u=(\int_{\mathbb{R}^n}K(y)u^{q}(y,t)dy)^{((p-1)/q)}u^{1+r},\quad & x\in \mathbb{R}^n,\quad t>0\\ u(x,0)=u_0(x)\quad & x\in \mathbb{R}^n \end{align*} where \(p>1\), \(q\geq 1\), \(r\geq 0\), \(u_0\in L^{\infty}(\mathbb{R}^n)\), \(u_0\geq 0\) and \(K\) is a positive, bounded continuos function. Section 46 is devoted to the following problem: \begin{align*} u_{t}-\Delta u=\int_0^{t}u^{p}(x,s)ds-ku^{q},\quad & x\in\Omega,\quad t>0\\ u=0,\quad & x\in\partial\Omega,\quad t>0\\ u(x,0)=u_0(x),\quad & x\in\Omega \end{align*} where \(\Omega\subset \mathbb{R}^n\) is a bounded domain, \(p>1\), \(q\geq 1\), \(k\geq 0\), \(u_0\in L^{\infty}(\Omega)\), \(u_0\geq 0\).The global existence and blow-up rate are studied.

Chapter 3 is devoted to elliptic and parabolic systems. In the case of elliptic systems (Section 31), the authors study the problem of a priori estimates and existence for weakly coupled systems. Section 32 deals with parabolic systems coupled by power source terms. Problems like well posedness, global existence and blow-up rate are studied here. In the last section (33), the authors study the different possible effects of adding linear diffusion to a system of ordinary differential equations. The results lead to the idea to consider some systems arising in biological or physical contexts such as mass-preserving and Grierer-Meinhardt systems. Chapter 4 is devoted to equations with gradient terms. The authors consider problems with nonlinearities depending on u and its space derivatives: \begin{align*} u_{t}-\Delta u=F(u,\nabla u),\quad & x\in\Omega,\quad t>0 u=0,\quad & x\in\partial\Omega,\quad t>0 u(x,0)=u_0(x),\quad & x\in\Omega \end{align*} where \(F=F(u,\xi):\mathbb{R}\times \mathbb{R}^n\to \mathbb{R}\) is \(C^1\)-function. In the Sections 36 to 39, \(F(u,\nabla u)=u^{p}+g(u,\nabla u)\) and only the nonnegative solutions are considered. In the Sections 40 and 41, the authors consider problems whose essential superlinear character comes from the gradient term. In these sections \(F(u,\nabla u)=|\nabla u|^{p}\), \(p>1\) in which case the problem arises in stochastic control theory, being also related with Kadar-Parisi-Zhang equation in the physical theory of growth and roughening of surfaces. The last chapter is devoted to various problems with nonlocal nonlinearities. The equations that are considered involve nonlocal terms taking the form of an integral in space or in time. In sections 43 and 44, the authors consider several equations with space integrals like: \[ u_{t}-\Delta u=\int_{\Omega}u^{p}(y,t)dy-ku^{q},\quad x\in\Omega,\quad t>0, \] where \(\Omega\subset \mathbb{R}^n\) is a bounded domain, \(p>1\), \(q\geq 1\), \(k\geq 0\), \(u_0\in L^{\infty}(\Omega)\), \(u_0\geq 0\), or \[ u_{t}-\Delta u=(\int_{\Omega}g(u)dx)^{m}f(u),\quad x\in\Omega,\quad t>0,\quad m\in \mathbb{R}^{\ast}. \] The global existence, blow-up and a priori estimates are studied. In Section 45, the authors deal with Cauchy problems with non local source terms involving space integrals of the form \begin{align*} u_{t}-\Delta u=(\int_{\mathbb{R}^n}K(y)u^{q}(y,t)dy)^{((p-1)/q)}u^{1+r},\quad & x\in \mathbb{R}^n,\quad t>0\\ u(x,0)=u_0(x)\quad & x\in \mathbb{R}^n \end{align*} where \(p>1\), \(q\geq 1\), \(r\geq 0\), \(u_0\in L^{\infty}(\mathbb{R}^n)\), \(u_0\geq 0\) and \(K\) is a positive, bounded continuos function. Section 46 is devoted to the following problem: \begin{align*} u_{t}-\Delta u=\int_0^{t}u^{p}(x,s)ds-ku^{q},\quad & x\in\Omega,\quad t>0\\ u=0,\quad & x\in\partial\Omega,\quad t>0\\ u(x,0)=u_0(x),\quad & x\in\Omega \end{align*} where \(\Omega\subset \mathbb{R}^n\) is a bounded domain, \(p>1\), \(q\geq 1\), \(k\geq 0\), \(u_0\in L^{\infty}(\Omega)\), \(u_0\geq 0\).The global existence and blow-up rate are studied.

Reviewer: Cristian Chifu (Cluj-Napoca)

### MSC:

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

35K51 | Initial-boundary value problems for second-order parabolic systems |

35K55 | Nonlinear parabolic equations |

35B44 | Blow-up in context of PDEs |

35J57 | Boundary value problems for second-order elliptic systems |

35J60 | Nonlinear elliptic equations |