The Quenching of Solutions of Linear Parabolic and Hyperbolic Equations with Nonlinear Boundary Conditions

In this paper we examine the initial-boundary value problems $(\alpha ):u_t = u_{xx}$, $0 \u3c x \u3c L,t \u3e 0$, $u(0,t) = u(x,0) = 0$, $u_x (L,t) = \phi (u(L,t))$ and $(\beta ):u_{tt} = u_{xx}$, $0 \u3c x \u3c L,t \u3e 0$, $u(0,t) = u(x,0) = u_t (x,0)$, $u_x (L,t) = \phi (u(L,t))$ where $\phi ( - \infty ,1) \to (0,\infty )$ is continuously differentiable, monotone increasing and $\lim _{u \to 1} - \phi (u) = + \infty$. For problem $(\alpha )$ we show that there is a positive number $L_0$ such that if $L \leq L_0$, $u(x,t) \leq 1 - \delta$ for some $\delta \u3e 0$ for all $t \u3e 0$, while if $L \u3e L_0 ,u(L,t)$ reaches one in finite time while $u_t (L,t)$ becomes unbounded in that time. For problem $(\beta )$ it is shown that if L is sufficiently small, then $u(L,t) \leq 1 - \delta$ for all $t \u3e 0$ while if L is sufficiently large and $\int_0^1 \phi (\eta )d\eta \u3c \infty ,u(L,t)$ reaches one in finite time whereas if $\int_0^1 \phi (\eta )d\eta = \infty ,u(L,t)$ reaches one in finite or infinite time. In either of the last two situations $u_t (L,t)$ becomes unbounded if the time interval is finite. If u reaches one in infinite time, then $\int_0^1 {u_x^2 } (x,t)dx$ and $u(x,t)$ are unbounded on the half line and half strip respectively

Numerical Solution of Ill Posed Problems in Partial Differential Equations

This project is concerned with several questions concerning the existence, uniqueness, continuous data dependence and numerical computation of solutions of ill posed problems in partial differential equations

The Role of Critical Exponents in Blowup Theorems

In this article various extensions of an old result of Fujita are considered for the initial value problem for the reaction-diffusion equation $u_t = \Delta u + u^p$ in $R^N$ with $p \u3e 1$ and nonnegative initial values. Fujita showed that if $1 \u3c p \u3c 1 + {2 / N}$, then the initial value problem had no nontrivial global solutions while if $p \u3e 1 + {2 / N}$, there were nontrivial global solutions. This paper discusses similar results for other geometries and other equations including a nonlinear wave equation and a nonlinear Schrödinger equation

Singularity Formation in Chemotaxis--A Conjecture of Nagai

Consider the initial-boundary value problem for the system (S)ut = uxx - (uvx)x, vt= u- av on an interval [0,1] for t \u3e 0, where a \u3e 0 with ux(0,t) = ux(1,t)= 0. Suppose \mu, v0 are positive constants. The corresponding spatially homogeneous global solution U(t) = \mu, V(t) = \mu a + (v0 - \mu a)\exp(-at) is stable in the sense that if (\mu\u27,v0\u27 ) are positive constants, the corresponding spatially homogeneous solution will be uniformly close to (U(\cdot),V(\cdot)). We consider, in sequence space, an approximate system (S\u27) which is related to (S) in the following sense: The chemotactic term (uvx)x is replaced by the inverse Fourier transform of the finite part of the convolution integral for the Fourier transform of (uvx)x. (Here the finite part of the convolution on the line at a point x of two functions, f,g, is defined as $\int_0^x(f(y)g(y-x)\,dy$.) We prove the following: If \mu \u3e a, then in every neighborhood of (\mu,v0 ) there are (spatially nonconstant) initial data for which the solution of problem (S\u27) blows up in finite time in the sense that the solution must leave L2 (0,1)\times H1 (0,1) in finite time T. Moreover, the solution components u(\cdot,t),v(\cdot,t) each leave L2 (0,1).If \mu \u3e a, then in every neighborhood of (\mu,v0 ) there are (spatially nonconstant) initial data for which the solution of problem (S) on (0,1) \times (0,Tmax ) must blow up in finite time in the sense that the coefficients of the cosine series for (u,v) become unbounded in the sequence product space $\ell^1\times\ell^1_1$. A consequence of (2) states that in every neighborhood of (\mu,v0 ), there are solutions of (S) which, if they are sufficiently regular, will blow up in finite time. (Nagai and Nakaki [Nonlinear Anal., 58 (2004), pp. 657--681] showed that for the original system such solutions are unstable in the sense that if \mu \u3e a, then in every neighborhood of (\mu,\mu a), there are spatially nonconstant solutions which blow up in finite or infinite time. They conjectured that the blow-up time must be finite.) Using a recent regularity result of Nagai and Nakaki, we prove this conjecture

Global Nonexistence Theorems for Quasilinear Evolution Equations with Dissipation

The final publication is available at Springer via https://doi.org/10.1007/s002050050032. Levine, Howard A., and James Serrin. "Global Nonexistence Theorems for Quasilinear Evolution Equations with Dissipation." Archive for Rational Mechanics and Analysis 137, no. 4 (1997): 341-361. Posted with permission.</p

Blow-up and pattern formation in hyperbolic models for chemotaxis in 1-D

In this paper we study finite time blow-up of solutions of a hyperbolic model for chemotaxis. Using appropriate scaling this hyperbolic model leads to a parabolic model as studied by Othmer and Stevens (1997) and Levine and Sleeman (1997). In the latter paper, explicit solutions which blow-up in finite time were constructed. Here, we adapt their method to construct a corresponding blow-up solution of the hyperbolic model. This construction enables us to compare the blow-up times of the corresponding models. We find that the hyperbolic blow-up is always later than the parabolic blow-up. Moreover, we show that solutions of the hyperbolic problem become negative near blow-up. We calculate the “zero-turning-rate” time explicitly and we show that this time can be either larger or smaller than the parabolic blow-up time. The blow-up models as discussed here and elsewhere are limiting cases of more realistic models for chemotaxis. At the end of the paper we discuss the relevance to biology and exhibit numerical solutions of more realistic models

The Quenching of Solutions of Some Nonlinear Parabolic Equations

[Mathematical equations cannot be displayed here, refer to PDF

The critical Fujita number for a semilinear heat equation in exterior domains with homogeneous Neumann boundary values

This article is published as Levine, H. A., and Q. S. Zhang. "The critical Fujita number for a semilinear heat equation in exterior domains with homogeneous Neumann boundary values." Proceedings of the Royal Society of Edinburgh Section A: Mathematics 130, no. 3 (2000): 591-602. DOI: 10.1017/S0308210500000317. Posted with permission.</p